IEEE MTT-V055-I07 (2007-07)

172 Pages·2007·16.93 MB·English

by  FANTOMASPING

JULY2007 VOLUME55 NUMBER7 IETMAB (ISSN0018-9480)
PAPERS
LinearandNonlinearDeviceModeling
SiGeHBT’sSmall-SignalPiModeling ...................................... T.-R.Yang,J.M.-L.Tsai,C.-L.Ho,andR.Hu 1417
Wideband Nonlinear Response of High-Temperature Superconducting Thin Films From Transmission-Line
Measurements .....................................................................J. Mateu,J.C.Booth,andB.H.Moeckly 1425
SmartAntennas,PhasedArrays,andRadars
Ultra-WidebandMultifunctionalCommunications/RadarSystem .............G.N. Saddik,R.S.Singh,andE.R.Brown 1431
AQuadratureRadarTopologyWithTxLeakageCancellerfor24-GHzRadarApplications ................................
.............................................................................................. C.-Y.Kim,J.-G.Kim,andS.Hong 1438
ActiveCircuits,SemiconductorDevices,andIntegratedCircuits
DesignofUltra-Low-VoltageRFFrontendsWithComplementaryCurrent-ReusedArchitectures..........................
........................................................................................................ H.-H.HsiehandL.-H.Lu 1445
ElectricalBackplaneEqualizationUsingProgrammableAnalogZerosandFoldedActiveInductors ......................
......................................................................................... J.Chen,F.Saibi,J.Lin,andK.Azadet 1459
MonolithicIntegrationofaFoldedDipoleAntennaWitha24-GHzReceiverinSiGeHBTTechnology...................
..............................E.Öjefors,E.Sönmez,S.Chartier,P.Lindberg,C.Schick,A.Rydberg,andH.Schumacher 1467
A4-bitCMOSPhaseShifterUsingDistributedActiveSwitches.................................. D.-W.KangandS.Hong 1476
FieldAnalysisandGuidesWaves
ANewBrillouinDispersionDiagramfor1-DPeriodicPrintedStructures ....................................................
................................................................... P.Baccarelli,S.Paulotto,D.R.Jackson,andA.A.Oliner 1484
ANonlinearFinite-ElementLeaky-WaveguideSolver ...................................P.C.AllilomesandG.A.Kyriacou 1496
EffectsofLossesontheCurrentSpectrumofaPrinted-CircuitLine............... J. Bernal,F.Mesa,andD.R.Jackson 1511
(ContentsContinuedonBackCover)
(ContentsContinuedfromFrontCover)
CADAlgorithmsandNumericalTechniques
MicrowaveCircuitDesignbyMeansofDirectDecompositionintheFinite-ElementMethod ..............................
..................................................................................................... V.delaRubiaandJ.Zapata 1520
FiltersandMultiplexers
Bandwidth-CompensationMethodforMiniaturizedParallelCoupled-LineFilters ...........................................
............................................................................................S.-S.Myoung,Y.Lee,andJ.-G.Yook 1531
MiniaturizedDual-ModeRingBandpassFiltersWithPatternedGroundPlane .......R.-J.Mao,X.-H.Tang,andF.Xiao 1539
Packaging,Interconnects,MCMs,Hybrids,andPassiveCircuitElements
DesignandHighPerformanceofaMicromachined -BandRectangularCoaxialCable....................................
....................................................................... M.J.Lancaster,J.Zhou,M.Ke,Y.Wang,andK.Jiang 1548
InstrumentationandMeasurementTechniques
ASwept-FrequencyMeasurementofComplexPermittivityandComplexPermeabilityofaColumnarSpecimenInserted
inaRectangularWaveguide ...................................................................................... A.Nishikata 1554
PhaseandAmplitudeNoiseAnalysisinMicrowaveOscillatorsUsingNodalHarmonicBalance...........................
........................................................................................... S.Sancho,A.Suárez,andF.Ramirez 1568
InformationforAuthors ............................................................................................................ 1584
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Boulder,CO80305USA DANIELDEZUTTER YOSHIO NIKAWA ZOYA POPOVIC RICHARD SNYDER
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AMIRMORTAZAWI KENJI ITOH JOSÉ PEDRO SANJAY RAMAN RUEY-BEEI WU
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USA USA
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IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.7,JULY2007 1417
SiGe HBT’s Small-Signal Pi Modeling
Tian-RenYang, JuliusMing-LinTsai, Chih-LongHo, and RobertHu
Abstract—This paper presents the derivation procedure used
indeterminingtheparametersinSiGeHBT’ssmall-signalmodel
wherethePicircuitconfigurationisemployed.Forboththetran-
sistor’s external base–collector capacitor and its base spreading
resistor, new close-form expressions have been derived. Com-
parisons with existing approaches vindicate the feasibility and
effectiveness of our formulations. With the impact of the lossy
substrate effectively modeled and the frequency dependency of
the transconductance properly addressed, this proposed extrac-
tion approach demonstrates accurate results up to 30 GHz with
differentbiasconditions.
IndexTerms—Basespreadingresistor,HBT,Pimodel,SiGe.
Fig. 1. HBT’s small-signal Pi model where the substrate network
consists of C , C , and R . Transconductance G is set to
I. INTRODUCTION G e =(1+j!(cid:28) ).
FOR THE small-signalmodeling ofan HBT,eitherTee or
Pi circuit configuration can be used [1]–[5]. Though the
Tee circuit reflects the device-physics aspect of this transistor,
the Pi circuit in general provides better insight into designing
circuits [6], [7] and, thus, will be explored in this paper. As
shown in Fig. 1, the SiGe HBT’s Pi model consists of the
intrinsic transistor, which is enclosed by the dotted box, the
basespreadingresistor ,thelossysubstratenetwork ,
,and [8]–[10],theexternalparasiticcapacitor ,
the base, emitter, and collector resistors , , and , and
theinputandoutputpadcapacitors and .Twotime
Fig.2. SiGeHBTundertest.(a)Photograph.(b)Schematicofthetransistor
constants and areaddedontothetransconductance to wheretheemitterisconnectedtoground,thebaseistheinput,andthecollector
accountforitsmagnitudeandphasefrequencydependency[5], istheoutput.
[11]–[15]. Output impedance of this voltage-induced current
sourceisassumedinfinite.Asiswellknown,onechallengein
layout consists of four fingers, with each 5.1- m long. Two-
SiGeHBT’ssmall-signalPimodelingcomesfromthepresence
portshort,open,load,andthru(SOLT)calibrationisperformed
of ,whoselocationbetween andtheintrinsictransistor
using 100- m Cascade probes on the ceramic substrate pro-
makes the reliable derivation of both and , so far,
videdbythesamevendor.WiththeAgilentnetworkanalyzer’s
largelybywayofadditionalteststructuresornumerically[3],
outputpowerset to 10dBm,losses duetothe additionalca-
[16]–[19].Inthispaper,properclose-formexpressionsforthese
blesandbias-Ts’willpullthepowerleveldownby0.2dB/GHz.
two parameters have been worked out and will be compared
DC bias for this transistor comes from HP4142B modular dc
withexistingapproaches[20]–[22].
source/monitor. The30-GHzupperfrequency ismainlydeter-
Fig.2showstheHBTundertest,whichisfabricatedusinga
mined by the available frequency range of the coaxial cables
commercial 0.35- m SiGe–BiCMOSprocessandhasbulkre-
usedinthemeasurement.
sistivityof8 cmforthesubstrate.Thebasepolyresistance
is200 square,whilethesilicidedbasepolyforinter-connec-
II. HBTSMALL-SIGNALPIMODELING
tionhasamuchlowerresistanceofafew square.Theemitter
A. Determinationof , , , ,and
ManuscriptreceivedOctober28,2006;revisedJanuary28,2007.Theworkof Fig. 3 shows the flowchart in determining the transistor’s
R.HuwassupportedbytheNationalScienceCouncil,R.O.C.,underContract
small-signalparameters. Tofindoutthe parasiticsofthe input
NSC95-2221-E-009-315.
T.-R.YangwaswiththeDepartmentofElectronicsEngineering,National and output pads, an open-pad test structure is designed where
ChiaoTungUniversity,Hsin-Chu,Taiwan300,R.O.C.,andalsowithVIATech- the transistor itself has been removed. Frequency-independent
nologies,Taipei,Taiwan100,R.O.C.HeisnowservingintheR.O.C.Army.
capacitance can, therefore, be obtained as fF,
J.M.-L.TsaiandC.-L.HoarewithVIATechnologies,Taipei,Taiwan100,
R.O.C. fF.The measuredcross-coupling capacitance be-
R.HuiswiththeDepartmentofElectronicsEngineering,NationalChiao tweeninputandoutputisthreeorderslessandcanbeneglected.
TungUniversity,Hsin-Chu,Taiwan300,R.O.C.(e-mail:shuihushuihu@yahoo.
Usinganappropriatedeembeddingprocedure,thesecapacitors
com).
DigitalObjectIdentifier10.1109/TMTT.2007.900214 can be removed from the transistor’s model. Since , ,
0018-9480/$25.00©2007IEEE
1418 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.7,JULY2007
Fig. 5. Reverse-biased HBT for the determination of substrate network.
(a)Schematic.(b)MeasuredandsimulatedsubstrateadmittanceY versus
frequency.Thesolidcurvesarethemeasuredresults;thedashedcurvesarethe
Fig. 3. Flowchart for the determination of the transistor’s small-signal simulatedoneswithC =21:4fF,C =57:5fF,andR =128(cid:10).
parameters.
thereverse-biasedintrinsictransistorresemblestwoseparateca-
pacitors,asshowninFig.5(a).Asfaras and arecon-
cerned, port 1 on the left of the schematic can be connected
to ground and the signal is injected into port 2 on the right. If
ismuchsmallerthantheimpedanceoftheseries cir-
cuit, then most of the current passing through from port 2
will flow down the branch rather than the branch.
By treating this as open circuit, we then have
,where
(1)
Fig.4. SaturatedHBTforthedeterminationofR ,R ,andR .(a)Schematic.
(b)Byextrapolatingthemeasuredresistancetothosecorrespondingtoinfinite
basecurrent,wehaveR =7:5(cid:10),R =4:1(cid:10),andR =4:3(cid:10).Thesolid Since the complex-number can provide only two
curvesaremeasuredat2GHz;thedashedonesareat5GHz.
constraints at each frequency point, analytical solutions (of
genuinely frequency independent) for the three parameters
constructing the substrate network cannot be obtained; rather,
and are beneath the first-layer metal, they are beyond the
numericalalgorithmsneedtobeused.As
reach of a short-circuit test structure, but can be determined
by forcing the transistor into saturation [23]. By setting the
current flowing out of the collector to be half of the base
(2)
current,weslowlyincreasethebasecurrentandvoltage,from
1 and 18.8 mV, respectively, to 11 and 95 mV, respectively.
Since this saturated intrinsic transistor can now be modeled
and
as two conducting diodes, as shown in Fig. 4(a), a Tee circuit
configuration,especiallyatlowfrequency,emerges.Byextrap-
olatingthemeasuredresistancetothatcorrespondingtoinfinite
base current, we have the frequency-independent , , and (3)
equal to 7.5, 4.1, and 4.3 , respectively, as illustrated in Least squares fit over the whole frequency range then gives
Fig. 4(b). Here, the solid curves are those corresponding to fF, fF,and ,respec-
2 GHz, and the dashed curves are for 5 GHz. The impacts of tively.Fig.5(b)showstheadmittanceofthesubstratenetwork.
both and on the transistor’s Pi model are ready to be Here,thesolidcurvesarethemeasuredresults;theoverlapping
removed now; , however, will be temporarily retained for dashedcurvesaretheirsimulatedcounterparts.Ofcourse,other
thedeterminationinSectionII-Bofthesubstratenetwork. similarformulationscanalsobeemployedforthederivationof
, ,and [9],[10].Ifonlyseries[24],orparallel,
circuitis usedtomodelthe substratenetwork, close-form
B. DeterminationofSubstrateNetwork
analytical expressions can indeed be written, but the resulting
Thoughmathematicallythesubstratenetworkcanbedecided parameters will be highly frequency dependent and, thus, are
when the transistor is in saturation, the small in-parallel notuseful.
renders the derived substrate parameter values highly suscep-
C. Determinationof
tibletomeasurementuncertainties.Reliableresultscanbeob-
tained by reverse-biasingthe transistor. With V, With both the substrate network and readily removed
A, V,and A, from the schematic of the reverse-biased transistor, analytical
YANGetal.:SiGeHBT’SSMALL-SIGNALPiMODELING 1419
Fig.6. Reverse-biasedHBTwherethesubstratenetworkandR inthepre-
viousschematichavebeendeembeddedforthepurposeofdeterminingC . Fig.7. Parametervaluesofthereverse-biasedtransistorwithdifferentbase
(a)Schematic.(b)MeasuredC versusfrequency. voltages.(a)R versusV wherethecirclemarkersarethederivedresults.
(b)C ,C ,andC versusV .
solutions for the remaining circuit components, as shown in
Fig.6(a),canbeobtainedonce and areknownwhere
(4)
Bydefining as
(5)
Fig.8. MeasuredandsimulatedS-parametersofthereverse-biasedHBT.The
we have
solidcurvesarethemeasuredresults;theoverlappingdashedcurvesaretheir
simulatedcounterparts.
Inthe conventional approachusing IC-CAP,detailed layout
information, dc capacitance measurement, and numerical fine
(6) tuninghavetobeemployedforthedeterminationof (and
). Besides, it postulates that has to be independent of
frequency. Recently, an analytical formulation for of a
If the admittance matrixof the , ,and sub-circuitis
normal-biasedtransistorhasbeenproposed[20].However,the
designatedas ,then
extensive use of least squares fits makes it to a large degree
resemble a numerical approach. In our case, at each fre-
(7) quencypointcanbedirectlycalculatedandcompared.Indeed,
asdemonstratedinSectionII-Don ,sometimesmorethan
and, thus, oneanalyticalsolutionscanbewrittenforacertainparameter;
however,notallofthemrenderthesameresultovertheintended
frequencyrange.
(8)
Figs. 8 and 9 show the -parameters of the reverse-biased
transistor. Here, the solid curves are the measured results; the
Themeasuredresultsare fF, , overlappingdashedcurvesaretheirsimulatedcounterparts.Bias
fF,andasshowninFig.6(b), fF.Thethreeother conditionandthecomponentvaluesusedinthesimulationare
expressionsderivedusing(7)alsogivethesamevalue.If tabulatedinTableI.
we change the base voltage while keeping the collector node
grounded,differentreverse-biasedparametervaluescanbeob-
tained,asillustratedinFig.7,whereboth and arein- D. DeterminationofNormal-Biased
dependentofthebasevoltageinthisreverse-biasedcondition.
Now,theimpactof onthetransistor’sPimodelcanbere- Now, with the bias of the transistor being set at V,
moved,i.e.,deembedded. mA(currentdensity1.13mA m ), V,and
1420 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.7,JULY2007
Fig.11. S and1=Y curvesusedinderivingR .(a)S usedfortheex-
trapolationof(16)from0.1to30GHz.(b)1=Y usedfortheextrapolationof
(17)from0.5to30GHz.Bothcurvesmoveclockwiseasfrequencyincreases.
Fig.9. Measured(solid)andsimulated(dashed)S-parametersofthereverse-
biasedHBTontheSmithchart.
On the other hand, if the transconductance can be treated as
a real number, we can also express in terms of and
TABLEI
REVERSE-BIASEDTRANSISTOR [21],i.e.,
(12)
with
(13)
and
(14)
Fig.10. R andthenormal-biasedintrinsictransistor.(a)Schematic.(b)Mea-
suredR wherecurve1isderivedusing(11),curve2isusing(12),andcurve
3isusingthealgorithmsuggestedin[20].
Here, the current flowing through needs to be assumed
much larger than that on the branch. Fig. 10(b) displays
A,thenormal-biased ,asshowninFig.10(a),
thederived wherecurve1comesfromourproposed(11),
canbedeterminedwhenboth and areknown.As
theupperboundcurve2isfrom(12),andthesomehowlower
boundcurve3isusingthealgorithmsuggestedin[20].Thedis-
crepancybetweenthesethreecurvesinstigatesafurthernumer-
icalsurveyasfollows.
AssuggestedintheIC-CAPuser’smanual,byassumingthe
transconductancetobearealnumber and
(9)
i.e.,
(10) (15)
then the input impedance (with 50- output loading) can be
therefore, expressedas
(16)
Extrapolation of the corresponding contour on the Smith
(11)
charttoinfinitefrequencyresultsina of25 ,asillustrate
inFig.11(a).Alternatively,byassuming tobemuchlarger
YANGetal.:SiGeHBT’SSMALL-SIGNALPiMODELING 1421
than the impedance of at high enough frequency, the in-
tended canbeobtainedbyextrapolatingthe curveto
the -axis[22],as
(17)
In Fig. 11(b), this is around 25 . Both numerical ap-
proaches,therefore,confirmouranalyticalmethod.
Regarding the derivation of , though the numerical ap-
proachesrendercorrectresults,theyareincapableofrevealing
thefrequencydependence(orindependence)of and,thus, Fig. 12. Measured and simulated transconductance. (a) Magnitude of
cannotbeviewedaswidebandmodelinginthisrespect.Strictly tYhe (cid:0)traYnsco;ntdhuectoavnecrelapwphinergedtahsehesdolcidurvceurivseitissstihmeulmateedasucroeudntererpsualrtt,wi.ieth.,
speaking, a valid exists only at infinite frequency. On G =123:8mS,(cid:28) =1:5ps,and(cid:28) =1:2ps.(b)Phaseofthetranscon-
the other hand, since the frequency variable is not used in the ductancewherethesolidcurveisthemeasuredresult;theoverlappingdashed
three discussed analytical (and, thus, wideband) methods, curve1isitssimulatedcounterpart;dashedcurve2isthesimulatedphasewith
only(cid:28) ,butno(cid:28) ;dashedcurve3iswith(cid:28) only.
at every frequency point can be independently obtained and
compared.Amongthethree,(12)hasanear-constant over
the widest bandwidth; however, the 6- offset relative to all
nonzero inthetransconductance,theconventional expres-
theotherdiscussedapproacheslimitsitsapplication.Whilethe
sionneedstoberevised.As
one suggested in [20] gives valid results for frequency from
15 to 30 GHz, our proposed (11) can have satisfying for
frequency from down below 10 to 30 GHz and, thus, is the
most preferred.
Furthermore,sincethis isdirectlyderivedattheintended
biaspoint,ratherthanadoptedfromvaluesusingotherbiascon-
ditions,thecurrentcrowdingeffect[25]–[28],evenifexists,will
notaffectthevalidityofourproposedformulation,andsincethe (19)
simulated -parametersagreewiththeirmeasuredcounterparts,
asdemonstratedinSectionII-E,itisjustfineusingasingle , where the leakage currents flowing through and are
ratherthanamorecomplicatedsub-circuit[29],inmodelingthis assumednegligibleathighfrequencyintheapproximation,we
partofthetransistor. then have
(20)
E. DeterminationofNormal-BiasedIntrinsicParameters
With deembedded, parameters of the normal-bi-
ased intrinsic transistor can, therefore, be determined as and
fF, , and fF. With the
transconductancedefinedas[5],[11]–[13] (21)
Fig.13showsthe curvesinlogarithmicandlinearscales.In
(18)
Fig.13(a),solidcurve1correspondstothetotaltransistor,solid
curve2iswiththeintrinsictransistoronly,thedashedstraight
where istheangularfrequency,wethenhave mS,
linesaretheirhigh-frequencylinearapproximationsinthislog-
ps,and ps.InFig.12(a),thesolidcurveis
arithmicscale.Fig.13(b)hasthesameresults,butexpressedin
themagnitudeofthemeasured andtheoverlappingdashed
linearscale.Thus, is42GHzwhenthetotaltransistorisem-
curveisitssimulatedcounterpart.Apparently, isneededfor
ployed,asbyextrapolatingthelogarithmiccurvetothe -axis,
explainingthismagnitudefrequencydependency.InFig.12(b),
andwehave equalto56GHzfortheintrinsicresistor,which
thesolidcurveisthephaseofthemeasured ,theoverlapping
agreeswiththe54.1GHzcalculatedusing(20);itis58.4GHz
dashedcurve1isitssimulatedcounterpartwithboth and
using(21),alsoavalidapproximation.Likewise,bydefiningthe
employed.Ifweretainthetimeconstant whileomitting ,
gain as[30]
or retain , but omitting , shown as dashed curves 2 and 3,
respectively,phasediscrepancycanbeobserved.
Knowingtheintrinsictransistor’sparametersnowenablesthe
(22)
calculation ofthe cutoff frequency ,which is the frequency
wherethemagnitudeofthecurrentgain isequalto1.With
1422 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.7,JULY2007
Fig.13. Magnitudeofh versusfrequency.(a)Inlogarithmicscalewhere
thesolidcurve1iswiththetotaltransistor;solidcurve2iswiththeintrinsic
transistoronly.Thetwodashedcurvesaretheirhigh-frequencylinearapproxi- Fig.16. Measured(solidline)andsimulated(dottedline)S-parametersofthe
mationforderivingf .(b)Inlinearscale. normal-biasedHBTontheSmithchart.
TABLEII
NORMAL-BIASEDTRANSISTOR
Fig.14. MagnitudeofgainUversusfrequency.(a)Inlogarithmicscalewhere
thetwooverlappingsolidcurvesarethemeasuredandsimulatedresultswith
thetotaltransistor;thedashedcurveisthehigh-frequencylinearapproximation
forderivingf .(b)Inlinearscale.
Fig.17. MeasuredandsimulatedS-parametersoftheHBTbiasedatV =2V,
I =6:6mA,V =0:95V,andI =65:7(cid:22)A.Thesolidcurvesarethemea-
suredresults;theoverlappingdashedcurvesaretheirsimulatedcounterparts.
Theerror(cid:15) is0.38%.
asshowninFigs.15and16.Here,thesolidcurvesarethemea-
suredresults;theoverlappingdashedcurvesaretheirsimulated
counterparts.Ifwedefinetheerror as
Fig.15. MeasuredandsimulatedS-parametersofthenormal-biasedHBT.The
solidcurvesarethemeasuredresults;theoverlappingdashedcurvesarethe
simulatedones.Theerror(cid:15) definedin(23)is0.13%.
(23)
where is the stability factor, the maximum oscillation fre-
quency ,whichisthefrequencyatwhich isequalto1, where the summation is over the frequency points
canbeeasilyobtained.Boththemeasured -parametersandthe from0.1to30GHz.Thecalculatederror is0.13%.Thebias
model-based -parameters give the same GHz,as conditionandparametervaluesusedinthesimulationaretabu-
showninFig.14. latedinTableII.
AccuracyoftheHBT’ssmall-signalPimodelingcanbever- Applyingthesamecollectorandbasevoltages,anothertran-
ifiedbycomparing themeasured andsimulated -parameters, sistor of the same size on the same wafer is measured, with
YANGetal.:SiGeHBT’SSMALL-SIGNALPiMODELING 1423
TABLEIII
TRANSISTORBIASEDATDIFFERENTBASEVOLTAGES
andthen0.73mA,withthecorrespondingcurrentdensitybeing
1.08,0.46,and0.12mA m ,respectively.Thesimulatedre-
sultsinthesecasesagreewiththeirrespectivemeasuredcoun-
terparts,asshowninFigs.18and19.Ontheotherhand,if is
changedfrom0.95to1.0V, willincreasefrom6.6to11.8mA
Fig.18. MeasuredandsimulatedS-parametersoftheHBTbiasedatV =2V, (currentdensity1.93mA m )withslightlyimprovedgainre-
I =2:8mA,V =0:90V,andI =19:4(cid:22)A.Thesolidcurvesarethemea- sponse at low frequency; the simulated results still follow the
suredresults;theoverlappingdashedcurvesaretheirsimulatedcounterparts.
measured ones, as shown in Fig. 20. Both and under
Theerror(cid:15) is0.27%.
thesedifferentbasevoltagesaretabulatedinTableIII.
III. CONCLUSION
In this paper, new procedures for deriving the SiGe HBT’s
small-signalPimodelinghavebeendeveloped.Fortheexternal
base–collector capacitor and the base spreading resistor
,reliableanalyticalsolutionshavebeenproposedandcom-
pared with other methods. The lossy substrate effect has also
beenappropriatelymodeled.Agreementsbetweenthemeasured
andsimulatedresultsineachderivationstepthusvindicatesthe
accuracyandefficiencyofournewmodelingapproach.Inaddi-
tiontotheintendednormal-biasedcondition,thisproposedap-
proachshowssatisfyingresultsfordifferentbiasingconditions.
Therefore, circuits designed using the HBT’s small-signal Pi
modelcanbeaccuratelyanalyzed.Inthefuture,weplantoex-
Fig.19. MeasuredandsimulatedS-parametersoftheHBTbiasedatV =2V, tendthismodelingworktoaddresseachparameter’snonlinear
I =0:73mA,V =0:85V,andI =4:6(cid:22)A.Thesolidcurvesarethemea- effect,thusfacilitatingthedesignofmixers.
suredresults;theoverlappingdashedcurvesaretheirsimulatedcounterparts.
Theerror(cid:15) is0.86%.
ACKNOWLEDGMENT
Theauthorswouldliketothanktheanonymousreviewersof
thisTRANSACTIONSforsuggestionsandencouragement.
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