IEEE MTT-V057-I10 (2009-10A)

193 Pages·2009·25.556 MB·English

by  FANTOMASPING

OCTOBER2009 VOLUME57 NUMBER10 IETMAB (ISSN0018-9480)
PART I OF TWO PARTS
PAPERS
LinearandNonlinearDeviceModeling
EfficientScalableModelingofDouble- EquivalentCircuitforOn-ChipSpiralInductors..................................
.............................................................................. Y.-G.Ahn,S.-K.Kim,J.-H.Chun,andB.-S.Kim 2289
SmartAntennas,PhasedArrays,andRadars
High-PerformanceIQModulator-BasedPhaseConjugatorforModularRetrodirectiveAntennaArrayImplementation ..
.................................................................................................... V.FuscoandN.B.Buchanan 2301
ActiveCircuits,SemiconductorDevices,andICs
AnExperimentalUltra-Low-VoltageDemodulatorin0.18- mCMOS ........................................................
.............................................................................. L.-S.Lai,H.-H.Hsieh,P.-S.Weng,andL.-H.Lu 2307
A1–20-GHzBroadbandMMICDemodulatorforLowIFReceiversinMultistandardApplications .......................
............................................................................................ C.Mohamed,A.Khy,andB.Huyart 2318
A12-GHz High-EfficiencyTaperedTraveling-WavePowerAmplifier WithNovelPowerMatchedCascodeGain Cells
UsingSiGeHBTTransistors .........................................................B.Sewiolo,G.Fischer,andR.Weigel 2329
OptimumASKModulationSchemeforPassiveRFIDTagsUnderAntennaMismatchConditions.........................
......................................... Y.Xi,S.Kwon,H.Kim,H.Cho,M.Kim,S.Jung,C.-S.Park,J.Kim,andY.Yang 2337
SignalGeneration,FrequencyConversion,andControl
ReflectionSolitonOscillator ...................................................... O.O.Yildirim,D.S.Ricketts,andD.Ham 2344
Millimeter-WaveandTerahertzTechnologies
60-GHzIntegratedTransmitterDevelopmentin90-nmCMOS.................................................................
........................................................... D.Dawn,P.Sen,S.Sarkar,B.Perumana,S.Pinel,andJ.Laskar 2354
(ContentsContinuedonBackCover)
(ContentsContinuedfromFrontCover)
WirelessCommunicationSystems
A60%PAEWCDMAHandsetTransmitterAmplifier ..........................................................................
............. P.A.Warr,K.A.Morris,G.T.Watkins,T.R.Horseman,K.Takasuka,Y.Ueda,Y.Kobayashi,andS.Miya 2368
FieldAnalysisandGuidedWaves
ANewTypeofCoupledCavityInteractionCircuitforGyrotronTravelingWaveAmplifierApplications .................
................................................................................. J.Luo,Y.Luan,W.Guo,M.Zhu,andG.Yuan 2378
EquivalentCircuitsforGap-andCoax-ExcitedCircularPostsinRectangularWaveguide................ A.G.Williamson 2384
PerformanceEvaluationofaPassiveMillimeter-WaveImager .................................................................
................................. Y.Li,J.W.Archer,J.Tello,G.Rosolen,F.Ceccato,S.G.Hay,A.Hellicar,andY.J.Guo 2391
CutoffWavelengthsofEllipticalMetallicWaveguides .............. G.D.Tsogkas,J.A.Roumeliotis,andS.P.Savaidis 2406
CADAlgorithmsandNumericalTechniques
ReductionofNumericalDispersionof3-DHigherOrderAlternating-Direction-ImplicitFinite-DifferenceTime-Domain
MethodWithArtificialAnisotropy ........................................................ Y.Zhang,S.-w.Lü,andJ.Zhang 2416
FiltersandMultiplexers
AdvancedCouplingMatrixandAdmittanceFunctionSynthesisTechniquesforDissipativeMicrowaveFilters ..........
............................................................................................................ V.MiraftabandM.Yu 2429
Packaging,Interconnects,MCMs,Hybrids,andPassiveCircuitElements
DesignandImplementationofDC–20-GHzLumpedResistorMatchedLoadsforPlanarMicrowaveCircuits............
.............................................................B.López-Berrocal,J.de-Oliva-Rubio,andI.Molina-Fernández 2439
AccurateSynthesisofFour-LineInterdigitatedCoupler..................................... L.Han,K.Wu,andX.-P.Chen 2444
InstrumentationandMeasurementTechniques
MeasurementofPlanarSubstrateUniaxialAnisotropy ..............................................J.C.RautioandS.Arvas 2456
Biological,Imaging,andMedicalApplications
AnalysisofOn-BodyTransmissionMechanismandCharacteristicBasedonanElectromagneticFieldApproach .......
.......................................................................................... J.Wang,Y.Nishikawa,andT.Shibata 2464
LETTERS
Commentson“A2.17-dBNF5-GHz-BandMonolithicCMOSLNAWith10-mWDCPowerConsumption”..... J.He 2471
Authors’Reply .......................................................................................... S.-S.LuandH.-W.Chiu 2472
InformationforAuthors ............................................................................................................ 2474
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Editors-In-Chief AssociateEditors
DYLAN WILLIAMS DANIEL DE ZUTTER JEN-TSAI KUO MAURO MONGIARDO RICHARD SNYDER
NIST Universiteit Gent Nat.ChiaoTungUniv. Univ. of Perugia RSMicrowaveCompany
Boulder,CO80305USA Belgium Taiwan Italy USA
Phone:[phone] email:[email protected] email:[email protected] email:[email protected] email:[email protected]
Fax:[phone]
email:[email protected] WOLFGANG HEINRICH YOUNGWOOKWON JOSÉ PEDRO CHI WANG
Ferdinand-Braun-Institut(FBH) Seoul Nat. Univ. Univ. of Aveiro OrbitalSciencesCorp.
AMIRMORTAZAWI Germany Korea Portugal USA
Univ. of Michigan email:[email protected] email:[email protected] email:jcp.mtted.av.it.pt email:[email protected]
AnnArbor,MI48109-2122USA
Phone:[phone] WEI HONG JENSHAN LIN ZOYA POPOVIC KE-LI WU
Fax:[phone] Southeast Univ. Univ. of Florida Univ.ofColorado,Boulder ChineseUniv.ofHongKong
email:[email protected] China USA USA Hong Kong
email:[email protected] email:[email protected] email:[email protected] email:[email protected]
ROBERT W. JACKSON
Univ.ofMassachusetts,Amherst
USA
email:[email protected]
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IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.10,OCTOBER2009 2289
(cid:0)
Efficient Scalable Modeling of Double- Equivalent
Circuit for On-Chip Spiral Inductors
Young-GhyuAhn, Student Member, IEEE, Seong-Kyun Kim, Student Member, IEEE,
Jung-Hoon Chun,Member,IEEE,and Byung-SungKim,Member,IEEE
Abstract—This paper presents an efficient technique to gen-
erateascalabledouble- circuitmodelfor spiralinductors.The
double- equivalent circuit is widely used to build an inductor
librarybecauseofitshighaccuracyoverawidefrequencyrange.
However,directparameterextractionforthedouble- modeland
scalable fitting for each parameter are complicated due to the
largenumberofcircuitelements.Theproposedmethodperforms
a scalable modeling using a relatively simple single- model
and converts the model to the double- model according to the
physics-based conversion algorithm. For this purpose, physical Fig.1. Single(cid:0)inductormodelforskinandproximityeffects.
relations between the single- and double- equivalent circuits
foraspiralinductorhavebeenderived.Theproposedtechniqueis
verifiedby developinga scalable inductorlibraryusing0.13- m
CMOStechnology.
Index Terms—Double- equivalent circuit, inductor, scalable
equations,scalablemodeling,parameterextraction.
I. INTRODUCTION
WITH THE advances in CMOS technology, the silicon
Fig.2. Basicstructureofdouble-(cid:0)modelfortheinductor.
monolithic spiral inductor becomes a critical passive
componentintheCMOSRFintegratedcircuits(RFICs)[1],[2].
Hence,theaccuratescalablemodelforthespiralinductorisin-
Today, the double- types of models are widely used to
evitableforreliableintegratedcircuit(IC)implementationand
achievethewidebandmodelingaccuracy,asnotedin[9],which
designoptimization.
isimportantinRFICdesigntoensurethestabilityofthecircuit
There have been much research to improve the accuracy of
beyondtheoperatingfrequencyrange.However,itsparameter
Si-based spiral inductormodelingduringthe pastdecade.The
extraction and scalable modeling are complicated due to the
simplesingle- modelforthespiralinductorwasintroducedin
increaseofcircuitelements.Toreducetheburdenofparameter
[3]andthemodifiedsingle- modelconsideringskinandprox-
extraction, an asymmetric double- model considering the
imityeffectsshowninFig.1waspublished[4],[5].However,
asymmetry of the inductor layout was proposed with the pa-
the above-mentioned models could not reflect the decrease in
rameterextractionforthesingle- cellsthroughthepartitionof
the effective series resistance (ESR) at high frequency caused
the double- equivalent circuit [10]. However, it is somewhat
by the substrate coupling. Thus, the advanced single- model
difficulttofindtheoptimalpartitionratiosandtheirscalability
withacouplingnetworkinthesubstratewasproposed[6].Re-
for various dimensions of inductors because the assumption
cently,formillimeter-waveCMOSdesign,amoresophisticated
thattheinductanceisproportionaltothelengthoftheinductor
equivalent-circuitmodelwasalsopublished[7].Thedouble-
is doubtful. In fact, the inductance is affected by not only the
model with the distributed nature [8]–[10] in Fig. 2 and the
length,butthenumberofturnsofaninductor.
modelincludingtheeddycurrent[11],[12]werealsoreported
Scalablemodelingproceduresarecomposedofparameterex-
fortheexplanationofthesubstratecoupling.
tractionforvariousdimensionsofinductors,andscalablefitting
of each parameter as a function of physical design parameters
ManuscriptreceivedJanuary08,2009;revisedJune10,2009.Firstpublished suchasthenumberofturns,diameter,width,etc.Fortheparam-
August25,2009;currentversionpublishedOctober14,2009.Thisworkwas
eterextraction,numericaloptimizationusing -parametermea-
supportedbytheKoreaMinistryofCommerce,IndustryandEnergyunderthe
Next-GenerationGrowthEngineProject. surementdataisapopularapproach.Numericaloptimizationis
Y.-G.Ahn,S.-K.Kim,andB.-S.KimarewiththeRFMicroelectronicDe- relativelystraightforward,butitssolutionisnotuniqueingen-
signLaboratory,SchoolofInformationandCommunication,Sungkyunkwan
eral;itisdifficulttogetscalabilityfortheparametersextracted
University,Suwon,Gyeonggi-do440-746,Korea(e-mail:[email protected].
kr). withvariousdimensionsofinductors.Onthecontrary,theana-
J.-H. Chun is with the High-Speed Integrated Circuits and Systems lyticparameterextractiontechnique[10],[13]–[15]issuitable
(HICS) Laboratory, Department of Semiconductor Systems Engineering,
for scalable modeling because its solutions are expected to be
SungkyunkwanUniversity,Suwon,Gyeonggi-do440-746,Korea.
DigitalObjectIdentifier10.1109/TMTT.2009.2029026 uniqueandexhibitphysicallymeaningfulscalablefeatures.
0018-9480/$26.00©2009IEEE
2290 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.10,OCTOBER2009
II. CONVERSIONOFSINGLETODOUBLE MODEL
A. EquivalentCircuitandExperimentalSetup
Fig.4illustrates theschematicofthe asymmetricoctagonal
spiralinductorandthecorrespondingequivalentcircuit.Consid-
eringthedistributedeffects,wecanderivethedouble- model
as a series of the single- models. The inductor is subdivided
into outer and inner inductors with the same length to reduce
thenumberoftheunknowns,andeachpartitionedinductorcan
be modeled separately as a single- cell shown in Fig. 4(b).
The ladder circuit with and in parallel to
is needed to capture the increase of the series resistance
due to skin and proximity effects as the frequency increases.
represents the capacitance between the in-
ductorandthesiliconsubstrate,and and represent
thesubstratecapacitanceandresistance,respectively[18],[19].
Fig.3. Flowchartoftheproposedmodelingprocedure. intheseriesbranchistheoverlapcapacitancebetweenthe
spiralandtheunderpassmetalline[10].Theasymmetricparam-
etersoftheseriesandshuntbranchesrepresenttheasymmetry
Inadditiontothefastandaccurateparameterextraction,scal- oftheinductor.
ability of the extracted parameters is very important to avoid In addition to the equivalent-circuit parameters of the
laboriousiterativefittingprocedurestobuildaunifiedscalable single- model, a mutual coupling coefficient between the
model.Thesubstrate-coupledsingle- modelwithscalableex- outer and inner inductors should be introduced to produce the
pressionswasreportedin[16]andthescalabilityintheT-model total inductance, which is larger than the sum of two self-in-
was published in [17]. Usual scalable modeling requires labo- ductancesofthepartitionedinductors.
riousfittingproceduresanditisgettingsevereasthenumberof For experimental verification of the proposed technique,
theequivalent-circuitparametersincreases.Thescalableequa- octagonal spiral inductors with various dimensions were
tionfitting aswell asthe parameter extractioninthe double- fabricated using 0.13- m one-poly and eight-metal CMOS
model with many unknown parameters is especially quite dif- technology. The top metal thickness is 3.3 m and the oxide
ficult. Relatively, the single- model is more compact for ex- thickness between the spiral inductor and silicon substrate is
tractionandmorereliableforscalablefittingthanthedouble- about12.7 m.Themicrophotographofthefabricatedinductor
model, whereas less accurate for wideband application due to withaguardringanditsequivalentcircuitareshowninFig.5.
substratecouplingathighfrequencyanddistributedcharacter-
B. ConversionofSingletoDouble Model
isticsoflarge-sizedinductor.Therefore,itisnecessarytoaccel-
eratethescalablemodelingproceduremaintainingtheaccuracy To reduce the complexity of parameter extraction and scal-
ofthedouble- model. able expression in the double- model with more than 15
This paper proposes a fast development method for the unknown independent parameters, an algorithm to convert
double- scalablemodelforanon-chipspiralinductorthrough the single- model to the double- model will be introduced.
theconversionfromthesingle- todouble- model.Allparam- Based on the conversion algorithm, we can perform the rela-
eter extraction and fitting of scalable equations are performed tively simple parameter extraction and scalable modeling for
forthesingle- equivalentcircuit.Thesingle- parametersare thesingle- modelandachievethehigh-frequencyaccuracyof
thenconvertedtothedouble- parametersusingnewlyderived the double- model at the same time. At first, we propose the
conversion relations. Additionally, a new extraction technique method to determine the mutual inductance between and
of the single- model parameters is developed. The scalable and present the modified equivalent circuit to incorporate
equation of each parameter in the equivalent circuit will be the mutual inductance as an equivalent-circuit element for
presented and the validity of the proposed approach will be easeofcalculation.Theconversionrelationsofeachparameter
verifiedthroughthecomparisonbetweenthemeasurementand betweenthesingle- anddouble- modelsarethenexplained.
thesimulationforthemainfeaturesoftheinductorsuchasthe 1) DeterminationoftheSelf-InductanceandMutualInduc-
inductance, quality factor, ESR, and self-resonant frequency tance: The mutual inductance is newly introduced when the
(SRF). inductor is partitioned into two inductors, and it cannot be di-
The proposed modeling and verification procedure is sum- rectlydeterminedfromthesingle- equivalent-circuitparame-
marized in the flowchart of Fig. 3. This paper is organized ters.Therefore,wehavetodeviseamethodtofindthemutual
according to this flow. Section II discusses the conversion inductanceanditshouldbeapplicableregardlessoftheinductor
algorithm of the single- to double- model and Section III designparameterswithoutanadditionalfittingforthedouble-
describes the parameter extraction technique of the single- equivalentcircuitornumericalcalculationforthepartitionedin-
model. Section IV explains the scalable model expressions ductors.Additionally,theself-inductances and should
and Section V shows the model verification results. Finally, also be determined from the modeling results of the single-
SectionVIconcludesthispaper. equivalentcircuit.Eventhoughtheinductorispartitionedwith
AHNetal.:EFFICIENTSCALABLEMODELINGOFDOUBLE- EQUIVALENTCIRCUITFORON-CHIPSPIRALINDUCTORS 2291
Fig.4. (a)Schematicoftheoctagonalspiralinductorwithasymmetryandthesingle-(cid:0)circuitfortheparameterextraction.(b)Theinductorisassumedtobe
dividedintohalves,whichhavethesingle-(cid:0)structure.Thereiscouplingfactor(cid:0)betweentheinductances(cid:0)(cid:2) (cid:3)(cid:2) (cid:2)oftheouter/innerinductors.Theentire
circuitisthedouble-(cid:4)structure.
establishedandvalidforvariousdimensionsoftheinductorsas
afunctionofthegeometricparameters
(1)
where and aretheinnerandouterdiametersofthein-
ductorand isthenumberofturns.Theinductance isalso
a function of conductor width and spacing , but those pa-
rametersdonotaffectthesubdivisionoftheinductor.Tosim-
plifythederivation,weomit and inequationsfrom(1)–(4).
Theself-inductancesoftheouter/innerinductorscanthenbeex-
pressedusing(1)asfollows:
(2)
The number of turns , the outer diameter , and the
inner diameter for the outer/inner inductors should be
gaugedtocalculate(2).Therelationoftheinductancesamong
the entire inductor and outer/inner inductors can be expressed
as
(3)
where ismutualinductanceand isthecouplingcoefficient.
Fig.5. (a)Microphotographofainductorwith3.5turnsand120-(cid:5)minner As a result, we can write the mutual inductance using (1) as
diameter.(b)Equivalentcircuitforinductor. follows:
thesamelength,eachself-inductancecannotbesettothehalfof
(4)
thetotalinductanceduetothedifferenceofthenumberofturns
and diameters. This paper solves the self-inductance and mu- Detailed expressionof (1) andits validitywillbe discussed in
tualinductance ofthe two partitionedinductorsfrom the scal- SectionsIVandV.
ableinductancemodelingresultsofthesingle- equivalentcir- Calculation of the network parameters using the coupling
cuit.InSectionI,weexplainedthattheproposeddouble- mod- coefficient is somewhat cumbersome because it begins with
eling approach utilized the single- modeling results given in voltage and current relations instead of simple impedance
theformofthescalableequations.Therefore,atthisstage,we and admittance calculations. To simplify the derivation, it is
assumethattheunifiedscalablemodelexpression(1)forthese- better to use the equivalent circuit incorporating the mutual
ries inductance inthe single- equivalentcircuitisalready inductanceasanexplicitcircuitcomponent,asshowninFig.6.
2292 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.10,OCTOBER2009
Fig. 6. Equivalent circuit for self-inductance and mutual inductance in the
T-branch.
Fig.8. Equivalentcircuitoftheinductoratafewgigahertzfrequency.
(7.2)
(7.3)
3) Conversion Equations of the Shunt Elements: As the
frequency enters the gigahertz range, the substrate coupling
through begins to influence the characteristic of the in-
ductor, but the substrate capacitance is still negligible.
Therefore, the asymmetric double- equivalent circuit can be
simplified as shown in Fig. 8, where the series resistances are
also ignored because the series reactances are dominant. The
Fig.7. ComparisonoftheESRbetweenthemeasurementandthesimulation
inthesingle-(cid:0)model. imaginary parts of the shunt admittances and
inthesingle- equivalentcircuitinFig.4(a)are
given by
Transformation to the T-equivalent circuit is similar to the
method explained in [20]. The negative sign in the T branch
arises because the current reference directions are identical in
(8.1)
and .
2) ConversionEquationsoftheSeriesElements: Thoughthe
single- modelfailstomodeltheESR,definedas ,
(8.2)
at high frequencies, as reported in [6], it matches well to the
measureddataatlowfrequencieswherethesubstratecoupling
where and at a few
isnegligible,asshowninFig.7[16].Accordingly,theresistance
gigahertzfrequencies. and inthe
oftheseriesbranchinthesingle- equivalentcircuitshouldbe
double- equivalentcircuitinFig.8canalsobeexpressedas
equal to the sum of impedances in each series branch of the
double- equivalentcircuitatlowfrequenciesasfollows:
(5) (9.1)
The series resistances of the outer and inner inductors should
especiallyhavethesamevalueatdcbecausethelengthofeach (9.2)
inductor is identical. Therefore, the following relation can be
expressed: where and
atafewgigahertzfrequencies.
(6) From(8)and(9),theoxidecapacitancesinthedouble- model
canbeobtainedasfollows:
Enforcingcondition(6)onrelation(5),thefollowingrelations
areobtainedbetweenthesingle-anddouble- circuits: (10.1)
(7.1) (10.2)
AHNetal.:EFFICIENTSCALABLEMODELINGOFDOUBLE- EQUIVALENTCIRCUITFORON-CHIPSPIRALINDUCTORS 2293
The real parts of the shunt admittances and
inthesingle- equivalentcircuitcanbewritten
as
(11.1)
(11.2)
atafewgigahertzfrequencies. and
(cid:0)
inthedouble- equivalentcircuitarealsowrittenasfollows: Fig.9. Fittingof(cid:0)(cid:2)(cid:3) (cid:4)(cid:0)(cid:2) (cid:5)asaquadraticfunctionof(cid:3) basedonthemea-
surementfora5.5-turninductor.Theequationis(cid:4)(cid:6)(cid:5)(cid:3) (cid:6)(cid:6)(cid:3) .
III. PARAMETER EXTRACTION FOR SINGLE-
(12.1) EQUIVALENTCIRCUIT
Thissectionwillpresentanewmethodologyfortheparam-
eterextractioninthesingle- modelwiththeconsiderationof
(12.2) theskinandproximityeffectswithoutthenumericaloptimiza-
tion.Therehavebeenmuchresearchconcerningtheparameter
extractioninthesingle- model[10],[13],[15].However,we
where and are already determined using relations
haveconfrontedsomedifficultiestoapplythepreviousmethods
(2)–(4)and .Therefore,formulasof(12)
becauseofthenoiseofmeasureddataatlowfrequenciesorthe
have onlytwo unknowns and . Equating (11) and
differentdatapatternsthatarenotobservedinthecitedpapers
(12) and solving the quadratic equations, we can easilyobtain
or possibility of multiple solutions. Therefore, this paper pro-
the solutions for and in closed forms. However,
posesnewapproachesforthescalableparameterextractionsin
sincethedetailedexpressionistoolengthy,buttrivial,weomit
thesingle- model.
theexactexpressions.
Todeterminethesubstratecapacitances,wecanrefertothe
A. ParameterExtractionfortheSeriesBranch
elementary electromagnetic analysis result that the product of
and is constant according to relation (13), which
Theimaginarypartoftheimpedanceoftheseriesbranchin
onlyrequiresthematerialconstantsofsiliconregardlessofthe
thesingle- equivalentcircuitcanbeexpressedas
shapeanddimensionofinductors[21].Therefore,thesubstrate
capacitancescanbeobtainedifthe constantisdetermined
(14)
(13)
The total inductance can be extracted from the
Since the material constants are uncertain in practice, we use
divided by radian frequency at low frequency
the constantextractedfromthesingle- equivalent-circuit
asfollows:
modeling. A detailed methodwill be givenin SectionIII.The
use of (13) can also reduce the complication of the scalable
(15)
modelsfortheinductorswithvariousdimensions.
Throughtheaboveconversionrelations,thescalableexpres-
sionsoftheparametersinthedouble- modelcanbeobtained For the accuracy of the model, the inductance in the
rightaftertheparameterextractionandthescalablemodelingin laddercircuitshouldbetakenintoconsiderationincalculating
the single- model without any additional laboriousand com- thetotalinductance.
plicated modeling procedures. The accuracy of conversion re- in(5)canbeexpandedintopowerseriesatlow
lationswillbeconfirmedinSectionVbycomparingthemod- frequencyasfollows:
eling results with the measured data. In the proposed method,
theaccuracyofthedouble- scalablemodelisbasicallydeter-
mined by the accuracy of the single- model parameters and
conversionrelations.Therefore,theparameterextractionofthe (16)
single- modelisveryimportant.InSectionIII,wewillpresent
a new and reliable parameter extraction method for single- Fig.9plotsthemeasureddataof atlowfrequency
equivalentcircuit. along with the fitted quadratic function of that is
2294 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.10,OCTOBER2009
Fig.10. Comparisonbetween(cid:0)(cid:2)(cid:3)(cid:4)(cid:0)(cid:2) (cid:5)and(cid:0)(cid:2)(cid:3)(cid:0)(cid:4)(cid:0)(cid:2) (cid:5)fortheselection Fig.11. (cid:6)(cid:7)(cid:3)(cid:0)(cid:4)(cid:0)(cid:8)(cid:2) (cid:9)(cid:2) (cid:10)(cid:5) asalinearfunctionof(cid:4) basedonthemea-
of the frequency range to extract the parameters in the single-(cid:3) circuit (5.5 surementfora5.5-turninductor.Theslopeistheoxidecapacitance.
turn).Thefrequencycriterionforthedataacquisitionisthefrequencywhere
(cid:0)
(cid:0)(cid:2)(cid:3) (cid:4)(cid:0)(cid:2) (cid:5)(cid:0)(cid:0)(cid:2)(cid:3)(cid:4)(cid:0)(cid:2) (cid:5)isabout95%.
.Eachparametercanbeobtainedfrom(16)and
thefittedquadraticequationasinthefollowing:
(17.1)
(17.2)
(17.3)
can also be obtained from (15) and (17). As shown in Fig.12. (cid:0)(cid:2)(cid:3)(cid:4)(cid:0)(cid:8)(cid:2) (cid:9)(cid:2) (cid:10)(cid:5)asafunctionof(cid:4) basedonthemeasurementfor
a5.5-turninductor.
Fig. 9, increases at first with the increase of
frequency due to the skin and proximity effects, but it begins
to decrease at high frequency due to the substrate coupling. Equation (18) can be expanded into power series at low fre-
of the single- equivalent circuit is not affected by the quencyasfollows:
shunt impedance, but of the actual inductor is influenced
asthefrequencyincreasesduetothedistributedeffects,which (19)
are incorporated in the double- model. Therefore, in the
quadraticfitting,itisimportanttolimitthehighendofthevalid
The measured data and fitting curve using (19) are plotted in
fitting range for . The criterion for the quadratic
Fig. 12 as a function of below the resonance frequency of
fittingrangefor canbedeterminedbycomparing
. and can be determined using the y-cut and
and , as shown in Fig. 10, because the
slope of the fitting curve. However, as can be seen in Fig. 12,
deviation between two data is mainly caused by the substrate
uniqueness of the values can be doubtful since the measured
coupling. Through many repeated experiments, the maximum
dataiswidelyscattered.Accordingtothecurve-fittingresults,
fitting range is determinedas the frequency where the ratio of
thevalueof isrelativelystable,buttheslopeofthefitting
isabout95%.
curve is sensitive to the spreading of the data; therefore, the
scalabilityof comestobepoor.Insteadofobtainingavalue
B. ParameterExtractionfortheShuntBranchand of fromtheslopedeterminedforeachinductor,thiswork
adoptsanindirectapproach.AsexplainedinSectionII,the
Theoxidecapacitancesof and inFig.4(a)canbe
product of the substrate model can be regarded as a constant
extractedeasily fromthe slope of and
independentofthedevicedimensions.Hence,wedeterminethe
asafunctionof atlowfrequency,as
averagevalueofthe productsofallinductorsaftersingle-
showninFig.11[13].
parameterextraction.Thevalueof isthennewlyupdated
The real part of in the single- equivalent
basedontheaverage constantandpredetermined for
circuitinFig.4(a)canbewrittenas
allinductors.Thismethodalleviatesthecomplexityofmaking
thescalablemodelfor .
The overlap capacitance can be extracted from the res-
(18)
onance frequency of the imaginary part of , which can be
AHNetal.:EFFICIENTSCALABLEMODELINGOFDOUBLE- EQUIVALENTCIRCUITFORON-CHIPSPIRALINDUCTORS 2295
TABLEI
LISTOFEXTRACTEDMODELPARAMETERS:(a)INTHESINGLE-(cid:2)MODEL
AND(b)INTHEDOUBLE-(cid:2)MODEL
Fig.13. Equivalentcircuitoftheinductoraroundresonancefrequencyof(cid:0) .
IV. EQUATIONSFORSCALABLEMODELS
In this section, we will confirm whether the parameters ex-
tracted in above-mentioned manner are scalable and construct
thescalableequationsforeachparameter.
A. ScalableExpressionsintheSingle- Model
Fig.14. Comparisonoftheresonancefrequencywith(cid:0)(cid:2)(cid:3)(cid:0) (cid:4) (cid:5) (cid:6)between
themeasurementandsimulationfora5.5-turninductor. Theparametersofthesingle- modelextractedusingthepro-
posedmethodarelistedinTableI(a).Forthescalablemodeling,
themonotonicityoftheextractedparametersalongthechange
derivedfromthedouble- equivalentcircuitinFig.13athigh oftheinductordimensionsisimportant.Fig.15showsthesuc-
frequencyasfollows: cessful scalability of those parameters for the various dimen-
sionsofinductors.
The following monomial scalable expression for the induc-
tancewasreportedin[22]:
(20)
(22)
where
where and aretheouterandaveragediametersofthe
inductor, isthewidthoftheconductor, isthespacebetween
the conductors, and is the number of turns. To incorporate
the inductance of the feeding line (A) within a guard ring in
Fig.4(a),weaddaconstantterm to(22)andmakeascalable
inductanceexpressionof inthesingle- modelgivenby
At the resonance frequency, of with (23)
canbeexpressedas
The inductance in the ladder circuit has a similar ten-
dencyas ,asshowninFig.15(b).Therefore,modifying(23),
(21)
the scalable expression of in the single- model is listed in
Table II.
Fig.14showstheresonancebehaviorofthemeasuredandsim- The scalable equations for and can be made from the
ulated in decibel scale for a 5.5-turn inductor. The simu- scalabilityof and showninFig.15(c)and(d).Theo-
lated result fits the measurement data with high accuracy over retically,theresistanceof islinearlyproportionaltothe
thewidefrequencyrange. length of the inductor, but in reality, the extracted data shows