IEEE MTT-V055-I11 (2007-11)

200 Pages·2007·25.32 MB·English

by  FANTOMASPING

NOVEMBER2007 VOLUME55 NUMBER11 IETMAB (ISSN0018-9480)
PAPERS
LinearandNonlinearDeviceModeling
Mildly Nonquasi-Static Two-Port Device Model Extraction by Integrating Linearized Large-Signal Vector
Measurements .................................................................... A. Cidronali,C.Accillaro,andG.Manes 2277
BehavioralThermalModelingforMicrowavePowerAmplifierDesign .......................................................
......................................... J.Mazeau,R.Sommet,D.Caban-Chastas,E.Gatard,R.Quéré,andY.Mancuso 2290
ActiveCircuits,SemiconductorDevices,andIntegratedCircuits
3-DIntegrationof10-GHzFilterandCMOSReceiverFront-End .............................................................
............................... T.Choi,H.Sharifi,H.H.Sigmarsson,W.J.Chappell,S.Mohammadi,andL.P.B.Katehi 2298
DesignofCryogenicSiGeLow-NoiseAmplifiers .................................... S.Weinreb,J.C.Bardin,andH.Mani 2306
ANewCompactLoadNetworkforDohertyAmplifiersUsinganImperfectQuarter-WaveLine ...........................
............................... H.Park,J.Van,S.Jung,M.Kim,H.Cho,S.Kwon,J.Jeong,K.Lim,C.Park,andY.Yang 2313
Linearization of CMOS Broadband Power Amplifiers Through Combined Multigated Transistors and Capacitance
Compensation................................................................. C.Lu, A.-V.H.Pham,M.Shaw,andC.Saint 2320
Millimeter-WaveandTerahertzTechnologies
Demonstrationofa311-GHzFundamentalOscillatorUsingInPHBTTechnology ..........................................
....... V.Radisic,D.Sawdai,D.Scott,W.R.Deal,L.Dang,D.Li,J.Chen,A.Fung,L.Samoska,T.Gaier,andR.Lai 2329
FieldAnalysisandGuidedWaves
Some Properties of Generalized Scattering Matrix Representations for Metallic Waveguides With Periodic Dielectric
Loading................................................................................................. S.S¸ims¸ekandE.Topuz 2336
PowerTransferinaLargeParallelArrayofCoupledDielectricWaveguides .......................................J.S.Wei 2345
GeneralizedImpedanceBoundaryConditionforConductorModelinginSurfaceIntegralEquation .......................
.......................................................................................... Z.G.Qian,W.C.Chew,andR.Suaya 2354
Rigorous Mode-MatchingMethodofCircular toOff-CenterRectangularSide-CoupledWaveguideJunctionsforFilter
Applications ................................................................................................ J.ZhengandM.Yu 2365
(ContentsContinuedonBackCover)
(ContentsContinuedfromFrontCover)
CADAlgorithmsandNumericalTechniques
PassivityEnforcementWithRelativeErrorControl..........................................S.Grivet-TalociaandA.Ubolli 2374
AnEfficientTime-DomainSimulationMethodforMultirateRFNonlinearCircuits ....... J.F.OliveiraandJ.C.Pedro 2384
FiltersandMultiplexers
DesignofBandpassEllipticFiltersEmployingInductiveWindowsandDielectricObjects .............. F.J.PérezSoler,
M. Martínez Mendoza, F. D. Quesada Pereira, D. Cañete Rebenaque, A. Alvarez Melcon, and R. J. Cameron 2393
A25–75-MHzRFMEMSTunableFilter ..........................K.Entesari,K.Obeidat,A.R.Brown,andG.M.Rebeiz 2399
ADual-BandCoupled-LineBalunFilter............................................................ L.K.YeungandK.-L.Wu 2406
AMicrostripUltra-WidebandBandpassFilterWithCascadedBroadbandBandpassandBandstopFilters ................
..................................................................................................... C.-W.TangandM.-G.Chen 2412
Packaging,Interconnects,MCMs,Hybrids,andPassiveCircuitElements
AParallel-StripRingPowerDividerWithHighIsolationandArbitraryPower-DividingRatio ...... L.ChiuandQ.Xue 2419
Slow-WaveLineCouplerWithInterdigitalCapacitorLoading .... L.Li,F.Xu,K.Wu,S.Delprat,J.Ho,andM.Chaker 2427
ASymmetricalFour-PortMicrostripCouplerforCrossoverApplication............................. Y. ChenandS.-P.Yeo 2434
ModifiedWilkinsonPowerDividersforMillimeter-WaveIntegratedCircuits ................................................
................................................. S.Horst,R.Bairavasubramanian,M.M.Tentzeris,andJ.Papapolymerou 2439
InstrumentationandMeasurementTechniques
InverseSyntheticApertureSecondaryRadarConceptforPreciseWirelessPositioning .....................................
.................................................................................. M.Vossiek,A.Urban,S.Max,andP.Gulden 2447
Biological,Imaging,andMedicalApplications
UsingaprioriDatatoImprovetheReconstructionofSmallObjectsinMicrowaveTomography ..........................
....................................................................................................... A.FhagerandM.Persson 2454
AReal-TimeExposureSystemforElectrophysiologicalRecordinginBrainSlices ..........................................
................A.Paffi,M.Pellegrino,R.Beccherelli,F.Apollonio,M.Liberti,D.Platano,G.Aicardi,andG.D’Inzeo 2463
InformationforAuthors ............................................................................................................ 2472
IEEEMICROWAVETHEORYANDTECHNIQUESSOCIETY
TheMicrowaveTheoryandTechniquesSocietyisanorganization,withintheframeworkoftheIEEE,ofmemberswithprincipalprofessionalinterestsinthefieldofmicrowavetheory
andtechniques.AllmembersoftheIEEEareeligibleformembershipintheSocietyuponpaymentoftheannualSocietymembershipfeeof$14.00,plusanannualsubscriptionfee
of$20.00peryearforelectronicmediaonlyor$40.00peryearforelectronicandprintmedia.Forinformationonjoining,writetotheIEEEattheaddressbelow.Membercopiesof
Transactions/Journalsareforpersonaluseonly. ADMINISTRATIVECOMMITTEE
J.S.KENNEY, President J.MODELSKI, PresidentElect K.G.GARD, Secretary N.KOLIAS, Treasurer
L.BOGLIONI D.HARVEY L.KATEHI T.LEE A.MORTAZAWI B.PERLMAN W.SHIROMA K.VARIAN K.WU
S.M.EL-GHAZALY J.HAUSNER B.KIM J.LIN V.J.NAIR A.ROSEN R.SNYDER R.WEIGEL R.YORK
M.HARRIS K.ITOH N.KOLIAS
HonoraryLifeMembers DistinguishedLecturers PastPresidents
T.ITOH T.S.SAAD K.TOMIYASU G.BOECK B.KIM J.C.RAUTIO M.SHUR K.VARIAN(2006)
A.A.OLINER P.STAECKER L.YOUNG W.HOEFER J.LASKAR D.ROOT P.SIEGEL K.C.GUPTA(2005)
T.ITOH V.LUBECKE D.RYTTING A.SUAREZ R.J.TREW (2004)
MTT-SChapterChairs
Albuquerque:S.BIGELOW Foothill: C.ANTONIAK NorthJersey:K.DIXIT SouthAfrica:P.W.VANDERWALT
Atlanta: D.LEATHERWOOD France:P.EUDELINE NorthQueensland: A.TSAKISSIRIS SouthAustralia:H.HANSEN
Austria: R.WEIGEL Germany: W.HEINRICH NorthernNevada: B.S.RAWAT SouthBrazil: L.C.KRETLY
Baltimore: A.D.BROWN Greece:I.XANTHAKIS Norway:S.E.WHEATLEY SoutheasternMichigan: L.M.ANNEBERG
Beijing: Z.FENG HongKong: W.Y.TAM OrangeCounty:H.J.DELOSSANTOS SouthernAlberta: S.BOUMAIZA
Beijing,Nanjing: W.X.ZHANG Houston:J.T.WILLIAMS Oregon: T.RUTTAN Spain: L.FEHARO
Belarus: A.GUSINSKY Houston,CollegeStation: C.MADSEN Orlando:P.WAHID Springfield: P.R.SIQUEIRA
Benelux: D.V.-JANVIER Hungary: T.BERCELI Ottawa: Q.YE Sweden: A.RYDBERG
Brasilia: A.KLAUTAU,JR. Huntsville:H.G.SCHANTZ Philadelphia:J.NACHAMKIN Switzerland: J.HESSELBARTH
Buenaventura: C.SEABURY India/Calcutta:P.K.SAHA Phoenix:C.WEITZEL Syracuse: E.ARVAS
Buffalo: E.M.BALSER IndiaCouncil: K.S.CHARI Poland:M.P.MROZOWSKI Taipei: C.-S.LU
Bulgaria: K.ASPARUHOVA Israel:S.AUSTER Portugal: C.A.CARDOSOFERNANDES Thailand: M.KRAIRIKSH
CedarRapids/CentralIowa: D.JOHNSON JapanCouncil:Y.TAKAYAMA Princeton/CentralJersey:W.CURTICE/A.KATZ Toronto: G.V.ELEFTHERIADES
CentralNewEngland: K.ALAVI Kitchener-Waterloo:R.R.MANSOUR Queensland:A.ROBINSON Tucson: N.BURGESS/S.MORALES
Central&SouthItaly: S.MACI Lithuania:V.URBANAVICIUS RiodeJaneiro: J.R.BERGMANN Turkey: O.A.CIVI
CentralNo.Carolina:T.IVANOV LongIsland/NewYork:J.COLOTTI Rochester:S.M.CICCARELLLI/J.VENKATARAMAN TwinCities: M.J.GAWRONSKI
Chicago: Z.LUBIN LosAngeles,Coastal:A.SHARMA Romania:I.SIMA UK/RI: A.REZAZADEH
Cleveland: G.PONCHAK LosAngeles,Metro: J.WEILER Russia,Nizhny-Novgorod: Y.BELOV Ukraine,CentralKiev: Y.POPLAVKO
Columbus:F.TEIXEIRA Malaysia:Z.AWANG Russia,SaintPetersburg:M.SITNIKOVA Ukraine,East: A.A.KIRILENKO
Connecticut:C.BLAIR/R.ZEITLER Melbourne:J.KRALOVEC Russia,Moscow:V.KALOSHIN Ukraine,Rep.ofGeorgia:R.ZARIDZE
Croatia: Z.SIPUS Milwaukee:S.G.JOSHI Russia,Saratov-Penza: N.RYSKIN Ukraine,Vinnitsya:V.DUBOVOY
Czech/Slovakia: P.HAZDRA MohawkValley: P.RATAZZI SaintLouis:D.MACKE Ukraine,West: I.ISAYEV
Dallas: R.EYE MonterreyMexico: R.M.RODRIGUEZ-DAGNINO SanDiego: J.TWOMEY Venezuela: M.PETRIZZELLI
Dayton: A.TERZOUOLI, JR. Montreal:K.WU SantaClaraValley/SanFrancisco: J.J.SOWERS Victoria:A.MITCHELL
Denver: M.JANEZIC NewHampshire: T.PERKINS Seattle:K.POULSON VirginiaMountain: D.MILLER
EasternNo.Carolina: D.PALMER NewJerseyCoast: D.REYNOLDS SeoulCouncil: H.-Y.LEE WashingtonDC/NorthernVirginia:J.QIU
Egypt: I.A.SALEM NewSouthWales: G.TOWN Siberia,Novosibirsk: V.SHUBALOV Winnipeg: V.OKHMATOVSKI
Finland: T.KARTTAAVI NewZealand: J.MAZIERSKA Siberia,Tomsk: O.STUKACH Yugoslavia: B.MILOVANOVIC
FloridaWestCoast: K.O’CONNOR NorthItaly: G.GHIONE Singapore: O.B.LEONG
Editors-In-Chief
DYLAN WILLIAMS AssociateEditors
NIST
Boulder,CO80305USA DANIELDEZUTTER YOSHIO NIKAWA ZOYA POPOVIC RICHARD SNYDER
Phone:[phone] Universiteit Gent Kokushikan Univ. Univ.ofColorado,Boulder RSMicrowaveCompany
Fax:[phone] Belgium Japan USA USA
email:[email protected] email:[email protected] email:[email protected] email:[email protected] email:[email protected]
AMIRMORTAZAWI KENJI ITOH JOSÉ PEDRO SANJAY RAMAN RUEY-BEEI WU
Univ. of Michigan MitsubishiElectronics Univ. of Aveiro VirginiaPolytech.Inst.andStateUniv. NationalTaiwanUniv.
AnnArbor,MI48109-2122USA Japan Portugal USA Taiwan, R.O.C.
Phone:[phone] email:[email protected] email:jcp.mtted.av.it.pt email:[email protected] email:[email protected]
Fax:[phone] JENSHAN LIN ALEXANDERYAKOVLEV
email:[email protected] Univ. of Florida Univ. of Mississippi
USA USA
email:[email protected] email:[email protected]
M.GOLIO, Editor-in-Chief,IEEEMicrowaveMagazine G.E.PONCHAK, Editor-in-Chief,IEEEMicrowaveandWirelessComponentLetters T.LEE, WebMaster
IEEE Officers
LEAHH.JAMIESON, PresidentandCEO JOHNBAILLIEUL, VicePresident,PublicationServicesandProducts
LEWISTERMAN, President-Elect PEDRORAY, VicePresident,RegionalActivities
CELIAL.DESMOND, Secretary DONALDN.HEIRMAN, President,IEEEStandardsAssociation
DAVIDGREEN, Treasurer PETERSTAECKER, VicePresident,TechnicalActivities
MICHAELR.LIGHTNER, PastPresident JOHNMEREDITH, President,IEEE-USA
MOSHEKAM, VicePresident,EducationalActivities
STUARTA.LONG, Director,DivisionIV—ElectromagneticsandRadiation
IEEE Executive Staff
JEFFRYW.RAYNES, CAE,ExecutiveDirector&ChiefOperatingOfficer
BETSYDAVIS, SPHR,HumanResources MATTHEWLOEB, CorporateStrategy&Communications
ANTHONYDURNIAK, PublicationsActivities RICHARDD.SCHWARTZ, BusinessAdministration
JUDITHGORMAN, StandardsActivities CHRISBRANTLEY, IEEE-USA
CECELIAJANKOWSKI, RegionalActivities MARYWARD-CALLAN, TechnicalActivities
SALLYA.WASELIK, InformationTechnology
IEEE Periodicals
Transactions/JournalsDepartment
StaffDirector:FRANZAPPULLA
EditorialDirector: DAWNMELLEY ProductionDirector: PETERM.TUOHY
ManagingEditor:MONAMITTRA SeniorEditor:CHRISTINAM.REZES
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES(ISSN0018-9480)ispublishedmonthlybytheInstituteofElectricalandElectronicsEngineers,Inc.Responsibilityforthe
contentsrestsupontheauthorsandnotupontheIEEE,theSociety/Council,oritsmembers.IEEECorporateOffice:3ParkAvenue,17thFloor,NewYork,NY10016-5997.IEEEOperations
Center:445HoesLane,P.O.Box1331,Piscataway,NJ08855-1331.NJTelephone:[phone].Price/PublicationInformation:Individualcopies:IEEEMembers$20.00(firstcopy
only),nonmember$85.00percopy.(Note:Postageandhandlingchargenotincluded.)Memberandnonmembersubscriptionpricesavailableuponrequest.Availableinmicroficheandmicrofilm.
CopyrightandReprintPermissions:Abstractingispermittedwithcredittothesource.Librariesarepermittedtophotocopyforprivateuseofpatrons,providedtheper-copyfeeindicatedinthe
codeatthebottomofthefirstpageispaidthroughtheCopyrightClearanceCenter,222RosewoodDrive,Danvers,MA01923.Forallothercopying,reprint,orrepublicationpermission,write
toCopyrightsandPermissionsDepartment,IEEEPublicationsAdministration,445HoesLane,P.O.Box1331,Piscataway,NJ08855-1331.Copyright©2007byTheInstituteofElectrical
andElectronicsEngineers,Inc.Allrightsreserved.PeriodicalsPostagePaidatNewYork,NYandatadditionalmailingoffices.Postmaster:SendaddresschangestoIEEETRANSACTIONS
ONMICROWAVETHEORYANDTECHNIQUES,IEEE,445HoesLane,P.O.Box1331,Piscataway,NJ08855-1331.GSTRegistrationNo.125634188.CPCSalesAgreement#40013087.Return
undeliverableCanadaaddressesto:PitneyBowesIMEX,P.O.Box4332,StantonRd.,Toronto,ONM5W3J4,Canada.
DigitalObjectIdentifier10.1109/TMTT.2007.911654
IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.11,NOVEMBER2007 2277
Mildly Nonquasi-Static Two-Port Device Model
Extraction by Integrating Linearized
Large-Signal Vector Measurements
AlessandroCidronali,Member,IEEE, CarmeloAccillaro,and GianfrancoManes,SeniorMember,IEEE
Abstract—Thispaperintroducesanewprocedure,basedonlin- large-signalvectormeasurementsattractsanumberofresearch
earized large-signal vector measurements, for extracting a non- groups, which are proposing significant approaches, and it is
linearbehavioralmodelfortwo-portactivemicrowavedevices.The still challenging for a number of other reasons. Amongst the
techniqueisappliedtoamodelstructurethatassumesashort-term
most important are: 1) the complete understanding of new
memory condition and is formulated as a parallel connection of
model formulations; 2) the new challenges arising from inno-
alimitednumberoffrequency-weightedstaticnonlinearities.The
vative metrology concepts; and 3) the problems related to the
proposed method consists of integrating the time-varying linear
characterization of the device driven into a nonlinear state by a newexperimentdesigns.
large signal. The experiment design and measurement setup are Thispaperisintendedtogiveacontributiontothesetopicsby
basedonalarge-signalnetworkanalyzerandarediscussedinde- introducinganewextractionprocedureforamildlynonquasi-
tail.Inthesecondportionofthispaper,insightisprovidedonthe static nonlinear model based on large-signal vector measure-
mostmeaningfulmodelparameters,alongwithanextensiveinde- ments. The technique proposed herein generalizes approaches
pendentexperimentalvalidation,whichconsidersaGaAspHEMT
basedontheintegrationofsmall-signalparametersinorderto
asacasestudyandincludestwo-tonelarge-signaldata,awideband
extractanonlinearmodel[11],[12].Thisisdonebyintroducing
codedivisionmultipleaccesssignal,bias-dependent -parameters,
amethodbasedonlinearizedlarge-signalparameterintegration.
anddcdata.
Incomparisonwithsimilarapproaches,theuseoflinearized
Index Terms—Behavioral modeling, computer-aided design
large-signalvectormeasurementsallowstheremovalofsignif-
(CAD) models, large-signal vector measurements, measure-
icant limitations that derive from an identification procedure
ment-basedmodels,nonlineardynamicmicrowavesystems.
based only on small-signal data and makes it possible to im-
provethemodelpredictionoflarge-signalstates.
I. INTRODUCTION The linearization of a large-signal device state was adopted
in[13]toacceleratetheconvergenceoftheVolterraserieseven
BEHAVIORAL modeling of microwave devices is experi-
for strongly nonlinear systems driven by large signals under
encinganincreaseofinterestbecauseofitscharacteristics
the hypothesis of short-term memory, while in [14], a similar
ofeffectiveness,technologyindependence,andeasyimplemen-
formulation is exploited in the complex-envelope domain for
tationinconventionalcomputer-aideddesign(CAD)tools[1],
bandpass nonlinear systems. In [13], the model identification
[2].Abehavioralmodelisgenerallydescribedintermsofstate
procedurereliesonpulsedI/Vandsmall-signalmeasurements.
functions, which are vector multiple-input functions of state
Extractionmethodsbasedonlinearizedvectormeasurementsin
variables. Their formulations are obtained by simplifying hy-
thecarrierdomainwereadoptedin[7]tocreateafrequency-do-
pothesesaboutcomplexdevicephysicsandtheyarecommonly
main behavioral model. The essential point in [7] was that, at
extractedbyprocessingvectorsmall-signal,scalarlarge-signal,
the system input, the only large signal was the fundamental,
and dc measurements [3]. In order to provide a meaningful
whereastheharmonicsareconsideredrelativelysmall.Insuch
interpolativecapability,behavioralmodelsshouldbeextracted
a condition, only the fundamentals at the incident input and
involving a broad range of possible device states, and usually
scattered output waves are large, while their harmonics are
theirpredictivecapabilityisamatterofconcern.Overthelast
small, hence subjected to a superposition principle. With that
decade, the development of new measurement techniques for
basicassumption,themodeliscurrentlylimitedtoasingle-tone
the vector analysis of devices under the nonlinear regime [4],
input, althoughit iscapableofconsidering modulatedcarriers
[5] has fostered new methods for identifying and validating
andtakingtheoutputmismatchintoaccountonlylinearly.The
such a class of models [6]–[10]. These methods demonstrated
approach presented herein is not subjected to the above-men-
improvementsinaccuratelypredictingthedevices’large-signal
tioned restrictions. In particular, the integration of linearized
dynamic nonlinear states. Behavioral modeling based on
large-signal parameters makes the adopted model capable of
predicting large-signal mismatches and responses to digitally
modulatedexcitation.Ontheotherhand,themethodintroduced
ManuscriptreceivedAugust10,2006;revisedJuly13,2007.Thisworkwas in [7] appears to be applicable to a wide range of devices and
supportedbytheInformationSocietyTechnologiesProgramoftheEuropean
circuit classes. Linearized large-signal vector measurements
UnionunderContractIST-1-507893-NOE.
The authors are with the Department of Electronics and Telecom- were also used in [10] for the identification of large-signal
munications, University of Florence, Florence I-50139, Italy (e-mail: scatteringfunctionsformodelingweaklynonlineardevicesand
alessandro.cidronali@unifi.it;carmelo.accillaro@unifi.it;gfm@unifi.it). signalsources.Incontrastto[10],themethodproposedherein
Colorversionsofoneormoreofthefiguresinthispaperareavailableonline
is capable of predicting higher order nonlinear behaviors that
athttp://ieeexplore.ieee.org.
areclearlynotpredictablebyusinglinearizedmodels.
DigitalObjectIdentifier10.1109/TMTT.2007.907379
0018-9480/$25.00©2007IEEE
2278 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.11,NOVEMBER2007
This paper is organized as follows. In Section II, the model where ,whiletheterm dependson
formulation is shown in terms of frequency-weighted parallel the thtimederivative.Thisnonlinearmodelisfrequentlycon-
nonlinear functions. The extraction procedure by a linearized structed from multibias small-signal parameters taken over a
large-signal characterization, along with the excitation design, wide range of frequencies and measured on a narrow voltage
arediscussedinSectionIII.SectionIVdescribestheindepen- grid,coveringthecompleteaccessiblebiasrangeoratleastthe
dent experimental validation by using two-tone large-signal range of interest. In [11], by comparing the measured small-
measurements, digitally modulated signals, bias-dependent signalbias-dependent -parameterswith(3),itisshownhowto
-parameters,anddccharacteristics.Finally,intheAppendix, approximate by best fitting the bias-dependent pseudoconduc-
amathematicaltreatmentdemonstratestheconsistencybetween tances whose highest order (i.e., the th nonlinear func-
the model formulation adopted in this paper and a previous tion)isdeterminedbythefrequencydispersionofthedevice’s
modelingapproach[13],[14]. performance.Thenonlinearfunctionsarethencalculatedbyin-
tegrating the bias-dependent into the 2-D voltage do-
main. The above-described approach restricts the range of the
II. MODEL EXTRACTION BASICS: INTEGRATING modelapplicabilitytoquasi-staticnonlinearsystems,although
THECONVERSIONMATRIX the higher order of the derivatives takes the frequency disper-
sion behavior into account. To achieve nonlinear models with
Wecanassumethatthecurrentattheterminalofadynamical
nonquasi-static capability, the extraction techniques are those
nonlinearsystemcouldbewrittenintheform
introducedin[12]and[13].
Incontrasttotheabove-describedtechniques,theextraction
methodintroducedinthispaperconsidersthenonlinearmodel
(1) formulated as in (1), but in (2), the partial differential deriva-
tivesaretakenatalarge-signalstate,1ratherthaninthequies-
cent point . This new formulation makes it possible to con-
where is the instantaneous input excitation. Equation (1)
sider, during the approximation of the model parameters, the
represents a general description of the behavioral model for
actual nonlinear behavior of the device-under-test (DUT), re-
dynamic nonlinear systems, and the number of terms adopted
sultingfromtheparticularexcitinglarge-signalwaveforms.The
intheformulationisdictated bythe memoryofthesystem. A
integration of the linearized large-signal parameters over the
formal mathematical treatment [15], given in the Appendix,
domain of interest makes the model in (1) capable of taking
showsitsderivationfromawidelyacceptedmodelingapproach
into account the nonquasi-static phenomena at microwave op-
[13]. The nonlinear functions are time invariant and
eration since a short, but finite, memory time is assumed (cf.
theirtimedependenceisby ,whilethehigherorderderiva-
theAppendix).Thisleadstoa“mildly”nonquasi-staticmodel,
tives allow accurate frequency-dependence representation of
whichcangiveaccuratepredictionofthedevicebehaviorwhen
the system response. In a conventional approach, as in [11],
excitedbymicrowavemodulatedsignals.
the identification procedure of can be obtained by
Next, the extraction procedure is described in detail. It is
integratingthebias-dependentsmall-signalresponse,which,in
centered on the capability of measuring the conversion matrix
thetimedomain,canbewrittenas
[16]oftheDUT,whichactuallyrepresentstheDUTlinearized
large-signalcharacterization.
ConsideringtheDUTdrivenbythelarge-signal, atthe
frequency ,thelinearizedcurrentresponsetoasmall-signal
excitation, atthefrequency ,is
(2)
where is the dc-bias point. Equation (2) is achieved by ex-
panding the current about the bias point with the Taylor se-
riesandretainingonlythefirstterm,whichisactuallytheone
responsible for the linearization. Under the basic assumption
thatthenonlinearfunctionsarecontinuous,thetimederivative
and the voltage derivative can be exchanged. Subtracting the (4)
large-signal current then leads to the small-signal incremental
currentin(2),inwhichthenumberoftermscorrespondstothe
number of terms in (1). Rewriting (2) into the frequency do-
main leads to a small-signal excitation response at the where isthedevicetime-domainsmall-signalcurrent.In(4),
frequency thepartialderivativesare timevariantanddepend onthe non-
linear dynamics of the device and the large-signal waveform.
Followingthereasoningin[16],wecanmovetothefrequency
1Although,inprinciple,thelarge-signalstatecanbeagenericone,duringthe
modelextraction,weconsidertheonedeterminedbytheexcitationwithalarge
(3)
signalappliedsimultaneouslyatthetwoDUTports.
CIDRONALIetal.:MILDLYNONQUASI-STATICTWO-PORTDEVICEMODELEXTRACTION 2279
domain,expandingthesmall-signaltime-dependantcurrentand
voltageasa“pseudo-Fourier”series
(5)
(6)
Fig.1. Schematicrepresentationofthemodelbyvoltage-controlledcurrent
where with ,where is sourcesandfrequency-domaindefinedweightfunctionssuchascanbefound
thehighestharmonicconsideredofthelargesignal,and is in[17].
aconvenientoffsetfrequencysothat isincludedamongthe
’s.Inthedefinitionof ,thenegativefrequenciescorrespond
tothelowersideband,asexpressedin[16].Theexpansionofthe single-valued expression representing the voltage-dependent
thtermin(4)isaproperFourierseriesandisgivenby admittance . This is easily obtained by plotting the
map that expresses, instant by instant, the values of
versusthecorrespondingvaluesof ,i.e.,
(7)
(11)
where are real coefficients with .
Equating memberto memberfor eachfrequency in(4) results which returns a scalar single-valued function. Finally, the cal-
inamatrixformas culation of the nonlinear functions is obtained by the
integration
(8)
(12)
where and are column vectors containing small-signal
The DUT nonlinear model defined based on is repre-
mixingproductsbetween and atthefrequencies .In
sentedinFig.1.Theimplementationofsuchamodelisstraight-
(8), represents the conversion matrix associated with the
forwardinmostofthecommercialCADtools,e.g.,byusingthe
nonlinear function driven by the large-signal and,
symbolicallydefineddeviceavailableviaAgilentTechnologies’
finally, isadiagonalmatrixwhosecoefficientsaretheangular
AdvancedDesignSystem(ADS)[17].Eachnonlinearfunction
frequenciesfrom to .Itisinterestingtoobservethe
isweightedbyaproperfactor, inFig.1,whichcorresponds
resemblance between the scalar equation in (3) and its vector
tothetimederivativein(1)expressedinthefrequencydomain.
generalizationrepresentedin(8).Havingassumedthat
arerealtime-invariantscalarfunctions, hastheform
III. MODELEXTRACTION
Here, a procedure is introduced for characterizing the lin-
... ... ... (9) earized large-signal DUT model and for extending the above-
described treatment in order to extract the nonlinear dynamic
modelforatwo-portdevice.
where is defined in (7). Similar to the above discussed
A. ExcitationDesign
small-signal case, bycomparing the measured conversionma-
trixwith(8),itispossibletoderivethematrices bycurve Thefirststepinidentifyingthenonlinearfunctionsrelieson
fitting.Equation(8)canbeconvenientlyrewrittenbyseparating the extraction of the DUT conversion matrix. The method is
realandimaginarycontributionsofthecurrent based on the technique proposed in [18] and [19] and a large-
signalnetworkanalyzer(LSNA)-basedmeasurementsetup[5].
Thelatterisslightlymodifiedinordertoaccommodatethesi-
multaneous excitation by a large signal of both DUT ports 1
(10)
and 2, as demonstrated in Fig. 2. The large signals are gener-
ated by a single synthesizer, SYNTH 1 in Fig. 2, to guarantee
Assuming further that is the independent variable of thesameamplitude,frequency,andphaserelationbetweenthe
the model extraction procedure, and setting its phase to 0, large-signalexcitationatboththeDUTportsduringtheentire
the real and imaginary contributions written in (10) are as- experiment. A phase stretcher, not shown in Fig. 2, can be in-
sociated, respectively, to the real and imaginary parts of the serted in one of the two branches to make a particular phase
equivalent time-variant pseudoadmittance seen from the DUT differencebetweenthelarge-signalexcitationatthetwoports.
terminals.Theaboveconsiderationallowsthedefinitionofthe The described setup makes it possible to define experiments,
2280 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.11,NOVEMBER2007
Thus, to identify the linearized response, excitation with ran-
domized phase and successive regression analysis are consid-
ered,thusincreasingthenumberofmeasurementsrequired.
B. ExtractionoftheModelforTwo-PortDevices
Once extracted, the two-port conversion matrix for the real
componentsrelatesvoltagesandcurrentsintheform
(14)
where the small-signal current and voltage vectors contain
the measured phasors. For example, in the case of
and , the notations are and
, respectively. A similar relation
holds for the imaginary components. If we start considering
a generic minor of (14), by following (9) and (10), we can
imposetheidentitybetweenthemeasuredconversionmatrices
andtheiranalyticalapproximation
. . . (15)
. . .
. . .
Fig.2. Measurementsetupforthesimultaneousexcitationofports1and2with
thelargesignalandonesmallsignal. where is the maximum degree of frequency expansion in
(10)requiredforabestfitapproximationofthelinearizedlarge-
signalfrequencybehavior.In(15),theunknownterms can
whichconsistofmeasuringthecurrentandvoltagephasorsof be calculated by imposing the identity, diagonal by diagonal;
themixingproducts,inanaprioridefinedfrequencygrid.These e.g.,forthemaindiagonal,wehave
aretheresultsfromasimultaneouslarge-andsmall-signalexci-
tations,thelatterappliedbySYNTH2inFig.2,ateachmixing
frequencydefinedby andateachoftheDUTports.
The calibration procedure described in [4] and [5] was ap-
pliedtoobtainabsolutecalibrateddata[4]attheDUTsections . . . .
. . . .
definedinFig.2.Thecorrectioncoefficientsremainthesamein .. . . . .
.
thetwostatesofthenonreflectiveswitchshowninFig.2.
Oncetherequiredmeasureddataareacquired,allthevoltage
readingsaredeembeddedfromthesystemimpedance (nor-
mally 50 ) to obtain the independent variables and for
themodelextraction.Thisisdonebyusingtherelation
.
.
.
(13)
(16)
where and arethecorrectedspectralcomponentsofthemea-
suredvoltagesandcurrentsatboththedeviceports.Tothisend,
we focus our attention only on mixing products. Considering being the column vector in the left part of (16), comprised of
this simple pre-processing, and after having separated the real dataextractedfrommeasurementsasaforesaid;adirectevalua-
and imaginary parts of the current , we are ready to apply tionoftheunknowncoefficients isachievedbyusingthe
the method defined in [19] to the and bestfittingsolution
datasetstoextractthemeasuredconversionmatrices and
, respectively. It is worth noting that the excitation de- (17)
signdescribedhereinisverysimilartothatadoptedin[7]with
somesubstantialdifferences.In[7],theapproachreliesonthe where is the maindiagonal of the measured and
small-signalperturbationofthelarge-signalstateatexactlythe denotes the transpose. The accuracy and stability of the
samefrequenciesasthoseofthefundamentalanditsharmonics. solution is directly related to the numerical condition of
CIDRONALIetal.:MILDLYNONQUASI-STATICTWO-PORTDEVICEMODELEXTRACTION 2281
the matrix . There are two sources for the large con-
dition number, the highest harmonic considered, i.e., , and
thedegreeofthefrequencyexpansionin(15),i.e., .These-
lection of is suggested by the degree of the nonlinearities
involvedintheproblemandbythelevelofthelarge-signalex-
citation. As in a harmonic balance problem, a low number of
harmonics might lead to solution inaccuracy, while large
might lead to high condition number. The dependency of the
condition number upon the frequency expansion is similar; as
increases, the condition number increases as well. For ex-
ample,considering ,with ,theconditionnumber
is ,whilewith ,itis .Theabovecalculation
canberepeatedforallthediagonalsof and untilall
oftheunknownmatrixentriesareidentified;thisprocessleads Fig.3. Graphicalsuperpositionofthepoweroftheincidentvoltagewavescal-
culatedfora50-(cid:10)characteristicimpedance,appliedtoport1throughthenine
tothecompleteidentificationofthematrixin(14).
experimentsforthecaseNH =4.
Thenextstepsofthemodelextractionprocedureconsiderthe
evaluationofthetime-varyingpseudoconductancesby(7)and
where and are the initial conditions. Finally, we can
theevaluationofthevoltage-dependentadmittances,whichare
writethecurrentatport2as
nowcalculatedfromparametrizedmapping
(18) (22)
Thesameformulationholdsforthecurrentatport1.
Thisconcludesthedeterminationofthenonlinearstaticfunc-
Theabove2-Dequationsareapproximatedbyconsideringthis
tionsforatwo-portdevice.
setofseriesexpansions
IV. INDEPENDENTEXPERIMENTALVALIDATION
The independent experimental validation of the previously
described method, and the comparison between measured and
modeleddcdata, -parameters,large-signal,multitoneexcita-
tion,anddigitallymodulatedsignalarediscussedhere.
A. MeasurementParameterData
(19)
The selected device for the model extraction is a 0.2- m
GaAs pHEMT with GHz with 180 m of total gate
where andthepair areparametersofthebasisfunc- periphery; withreference to the measurement setup defined in
tions,while arerealcoefficientsresultingfromthebest Fig.2,ports1and2correspondtothegateanddrainterminals,
fitting procedure in the domain. The selection of the respectively.The set of parameters for the modelextractionis
basisfunctionsin(19)issuggestedbytheparticularshapeofthe thefollowing: GHzand MHz,threehar-
,which,underfurtherrestrictionsonthecoefficients monicsareconsideredandthemaximumfrequencyinvolvedin
theexperimentis19.8GHz.Theselectionof wasdictated
, leads to equal mixed partial first-order derivatives. In
bytheminimumspacingofthecalibratedfrequencygridofthe
thiscondition,theintegrationof ,e.g.,forport2,
LSNA. The bias points are V, while
spans from 1 to 5 V in steps of 1 V. The large-signal tone is
8.9dBmtoensuremaximumcoverageoftheregionofinterest
frompinchofftosaturation.Thesmallsignalis setconstantat
20dBmovertheentirefrequencyrange.
(20)
The graphical superposition of the measured power of the
becomes path independent. Since is an arbitrary path in the
incident voltage wave spectra at port 1 is given in Fig. 3 for
domain,theintegralsin(20)canbesolvedbyadopting
all nine experiments required in the current case of ,
thefollowingtwoorthogonalpaths,asin[12]:
i.e., ,whilethecorresponding
scatteredpowerofthevoltagewavespectraaregiveninFig.4.
Thesequantitiesarecalculatedhere,aswellashenceforth,from
the incident voltage wave and the scattered voltage wave ,
which are related to the signal port current and voltage in a
50- characteristicimpedancesystem,bytherelationships
(21) and ,respectively.Aspread
in the test signal amplitude is observed due to the loss in the
2282 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.11,NOVEMBER2007
Fig.4. Graphicalsuperpositionofthepowerofthescatteredvoltagewaves
calculatedfora50-(cid:10)characteristicimpedance,fromtheDUTport2asresponse
tothesignalshowninFig.3.
Fig.6. 3-DplotforY forVgate=(cid:0)0:45VandVdrain=3V.
TABLEI
MEASUREMENT UNCERTAINTY CALCULATED ON
MAGNITUDEOFa1(INFIG.3)ANDb2(INFIG.4)
Fig.7. 3-DplotforY forVgate=(cid:0)0:45VandVdrain=3V.
determines the model range of
validity.
B. ModelInsightView
Fig.5. Graphicalsuperpositionofthemeasureddcdevicecharacteristic(con-
tinuouscurves),Vgatespanningfrom+0.5to(cid:0)0.8V,andthemeasureddy- The bias-dependent conversion matrix was extracted by
namicloadlines,thelatterforaconstantVgate = (cid:0)0:45V andVdrain considering the measured data and the procedure described
spanningfrom1to5Vwithinputpowerof8.9dBm. in Section IV-A. In Figs. 6 and 7, the 3-D plot of the values
are given for the real and imaginary parts, respectively, of the
minor of the conversion matrix in (14) of the bias point
measurementsetup,whilerepeatingmeasurementsofthelarge V and V. In the above figures,
signalanditsharmonicsexhibitedthemeasurementuncertainty the main diagonals refer to the ratio between and with
due to random effects and to the application of the test signal spanning from 4 to 4 with all the with .
reportedinTableI.InTableI,themeanvalueandstandardun- Fig. 6 shows that the conversion matrix has the form of a
certaintyarereportedaccordingtotherecommendationsin[20]. band matrix with diagonals slowly varying in value. A more
In Fig. 5, the dynamic load lines described by the instanta- interesting aspect is that the coefficients with
neous bias point are described as the result of a large-signal to and are negative. This is directly associated
application to both device ports and a small signal at the with the third-order nonlinearity, which is responsible for the
lowest frequency at port 1 (namely, experiment #1 in Fig. 3). compression and, consequently, has a negative multiplying
In this same figure, the measured dc characteristic is also coefficient.Thediagonalsvanishmovingfromthemaintothe
reported. The dynamic load-line coverage in the phase space outer diagonals, demonstrating that, for this particular case,
CIDRONALIetal.:MILDLYNONQUASI-STATICTWO-PORTDEVICEMODELEXTRACTION 2283
Fig.8. Nonlinearfunctionsg (v ;v ).Squares:k=0.Circles:k=2.Tri- Fig.10. Simulated(crosses)andmeasured(squares)incidentsweepinginput
(cid:0)
angles:k=4.Diamonds:k=6.Crosses:k=8.g (v ;v )aremultiplied wave from 22 to 5 dBm for a two-tone signal at frequencies of 4.2 and
by(2(cid:25)f ) andv = 3V. 5.4GHz.
Fig.11. Simulated(crosses)andmeasured(squares)scatteredoutputwavefor
(cid:0)
thetwo-tonesignalinFig.9,biaspointVgate= 0:45VandVdrain=3V.
Fig.9. Nonlinearfunctionsg (v ;v ).Squares:k=1.Circles:k=3.Tri-
angles:k=5.Diamonds:k=7.Crosses:k=9.g (v ;v )aremultiplied
by(2(cid:25)f ) andv = 3V. Fornegativeinputvoltages,thedevice’scurrentcomponentsus-
tainedbythedynamiccurrentsbecomesrelevant.Thefunctions
areweaklynonlinear,mostlypositive,andvanishbyincreasing
three harmonics are enough to represent the entire nonlinear theirorder,i.e.,thevalueof .Thissamefigureshowsthatthe
behavior. selection of five static nonlinearities represents a good choice
Fig.7depictsamorerelevantmaindiagonalvariation.This for the device in the frequency range under consideration [we
typicalbehaviorisassociatedwiththeoutputparallelnonlinear canassessthatthesameistrueforeachoftheminorsin(14)].
capacitance;thenegativesignisnotsurprisingsincetheupper Thenonlinearfunctionsrelatedtotheimaginarypartsshownin
corner links the current and voltage complex conjugate, thus Fig. 9 are all multiplied by a factor with . All
justifyingtheminussign. the functions collapse in V as expected because they
The insight into the model ends with the discussion of the donotprovidecontributionatdc.Themaincontributionisthe
nonlinearfunctionsextractedbytheprocedureofintegratingthe onerelatedto ,purecapacitivebehavior.Thedevicere-
conversionmatrices.Inthecurrentcase,anumberofstaticnon- spondswithmorecomplexbehavior,whichisapproximatedby
linearities equaltotenwereselected,whichmeansthatin(1), thehigherorderfunctions.
and in (15), . Figs. 8 and 9 show the nonlinear
C. Two-ToneValidation
functionsrequiredtomodelthecurrentatport2.Moreprecisely,
they show only the contributions arising from and The large-signal validation starts by considering a two-tone
minorsoftheconversionmatrixin(14);thiscorrespondstothe signalappliedtoport1only, GHzand GHz
firstmemberintheright-handsideof(21)fortherealandimag- in the bias point V and V, and the
inary parts, respectively. In the previously mentioned figures, nominalinputpowerwassweptfrom 22to5dBm.Thesim-
foranhomogeneousrepresentation,thenonlinearfunctionsare ulatedandmeasuredincidentinputwavesareshowninFig.10;
multipliedbytheradialfrequencyadoptedduringthecharacter- thedifferencebetweenlevelsofthetwotonesisdeterminedby
ization,i.e.,4.8GHz,tothepowerof . thesetuplosses.Asresponsetothisinputsignal,thedeviceex-
InFig.8,thefunctioncorrespondingto isthedccontri- hibitedthescatteredoutputwaveillustratedinFig.11.Thehar-
butionandrepresentsthestatictransconductance;asexpected, monicgeneration,intermodulation,andaccuracyofthemodel
itistheonlycontributiontothecurrentatport2for V. inpredictionofsuchcontributionsisobservableinthisfigure.