IEEE MTT-V057-I08 (2009-08)

244 Pages·2009·30.846 MB·English

by  FANTOMASPING

AUGUST2009 VOLUME57 NUMBER8 IETMAB (ISSN0018-9480)
PAPERS
LinearandNonlinearDeviceModeling
AnalysisofSeries-ConnectedDiscreteJosephsonTransmissionLine .................... H.R.MohebbiandA.H.Majedi 1865
SmartAntennas,PhasedArrays,andRadars
AProgrammableLens-ArrayAntennaWithMonolithicallyIntegratedMEMSSwitches....................................
........................................................... C.-C.Cheng,B.Lakshminarayanan,andA.Abbaspour-Tamijani 1874
ActiveCircuits,SemiconductorDevices,andICs
AnalysisandCompensationofPhaseVariationsVersusGaininAmplifiersVerifiedbySiGeHBTCascodeRFIC .......
.............................................................................. F.Ellinger,U.Jörges,U.Mayer,andR.Eickhoff 1885
A40-GHzLow-NoiseAmplifierWithaPositive-FeedbackNetworkin0.18- mCMOS...... H.-H.HsiehandL.-H.Lu 1895
A22–29-GHzUWBPulse-RadarReceiverFront-Endin0.18- mCMOS .... V.Jain,S.Sundararaman,andP.Heydari 1903
CMOSActiveInductorLinearityImprovementUsingFeed-ForwardCurrentSourceTechnique ...........................
...................................................................................C.-L.Ler,A.K.B.A’ain,andA.V.Kordesch 1915
A90-WPeakPowerGaNOutphasingAmplifierWithOptimumInputSignalConditioning ................................
J.H.Qureshi,M.J.Pelk,M.Marchetti,W.C.E.Neo,J.R.Gajadharsing,M.P.vanderHeijden,andL.C.N.deVreede 1925
TheoryandExperimentalResultsofaClassFAB-CDohertyPowerAmplifier ..............................................
.........................................................................P.Colantonio,F.Giannini,R.Giofrè,andL.Piazzon 1936
SignalGeneration,FrequencyConversion,andControl
Design of 24-GHz 0.8-V 1.51-mW Coupling Current-Mode Injection-Locked Frequency Divider With Wide Locking
Range................................................................................Z.-D.Huang,C.-Y.Wu,andB.-C.Huang 1948
LowPhase-NoisePlanarOscillatorsEmployingElliptic-ResponseBandpassFilters.........................................
.............................................................................................J.Choi,M.Nick,andA.Mortazawi 1959
Analysis and Design of Reduced-Size Marchand Rat-Race Hybrid for Millimeter-Wave Compact Balanced Mixers in
130-nmCMOSProcess ............................ C.-H.Lien,C.-H.Wang,C.-S.Lin,P.-S.Wu,K.-Y.Lin,andH.Wang 1966
A5-GHzCMOSType-IIPLLWithLow andExtendedFine-TuningRange ........ S.P.BrussandR.R.Spencer 1978
(ContentsContinuedonBackCover)
(ContentsContinuedfromFrontCover)
FieldAnalysisandGuidedWaves
AnalysisofaPostDiscontinuityinanOversizedCircularWaveguide .........................................................
........................................................................S.B.Sharma,V.K.Singh,R.Dey,andS.Chakrabarty 1989
CharacterizationofthePropagationPropertiesoftheHalf-ModeSubstrateIntegratedWaveguide .........................
...............................................................................Q.Lai,C.Fumeaux,W.Hong,andR.Vahldieck 1996
CADAlgorithmsandNumericalTechniques
ImplicitElementClusteringforTetrahedralTransmission-LineModeling(TLM)............................................
......................... P.D.Sewell,T.M.Benson,C.C.Christopoulos,D.W.P.Thomas,A.Vukovic,andJ.G.Wykes 2005
A3-DRadialPointInterpolationMethodforMeshlessTime-DomainModeling........................ Y.YuandZ. Chen 2015
A Linear-Time Complex-Valued Eigenvalue Solver for Full-Wave Analysis of Large-Scale On-Chip Interconnect
Structures .................................................................. J.Lee,V.Balakrishnan,C.-K.Koh,andD.Jiao 2021
IncorporationofMultiportLumpedNetworksIntotheHybridTime-DomainFinite-ElementAnalysis ....................
............................................................................................................ R.WangandJ.-M.Jin 2030
Packaging,Interconnects,MCMs,Hybrids,andPassiveCircuitElements
RFDesign,PowerHandling,andHotSwitchingofWaveguideWater-BasedAbsorptiveSwitches.........................
......................................................................................................C.-H.ChenandD.Peroulis 2038
Design and Modeling of a Stopband-Enhanced EBG Structure Using Ground Surface Perturbation Lattice for
Power/GroundNoiseSuppression ................................ T.-K.Wang,C.-Y.Hsieh,H.-H.Chuang,andT.-L.Wu 2047
BroadbandLumped-ElementIntegrated -WayPowerDividersforVoltageStandards .....................................
............................ M.M.Elsbury,P.D.Dresselhaus,N.F.Bergren,C.J.Burroughs,S.P.Benz,andZ.Popovic´ 2055
OptimumDesignofWidebandCompensatedandUncompensatedMarchandBalunsWithStepTransformers ...........
........................................................................................................Z.XuandL.MacEachern 2064
Physics-BasedViaandTraceModelsforEfficientLinkSimulationonMultilayerStructuresUpto40GHz ..............
.............................................................................................................. R.Rimolo-Donadio,
X. Gu, Y. H. Kwark, M. B. Ritter, B. Archambeault, F. de Paulis, Y. Zhang, J. Fan, H.-D. Brüns, and C. Schuster 2072
Microwave Photonics
OpticalMillimeter-WaveUp-ConversionEmployingFrequencyQuadruplingWithoutOpticalFiltering ..................
..................................... C.-T.Lin,P.-T.Shih,J.Chen,W.-J.Jiang,S.-P.Dai,P.-C.Peng,Y.-L.Ho,andS.Chi 2084
SampledAnalogOpticalLinks ............................................................ J.D. McKinneyandK.J.Williams 2093
InformationforAuthors ............................................................................................................ 2100
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SpecialIssueonTHzTechnology:BridgingtheMicrowave-to-PhotonicsGap .............................................. 2101
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DYLAN WILLIAMS DANIEL DE ZUTTER JEN-TSAI KUO MAURO MONGIARDO RICHARD SNYDER
NIST Universiteit Gent Nat.ChiaoTungUniv. Univ. of Perugia RSMicrowaveCompany
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Univ.ofMassachusetts,Amherst
USA
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IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.8,AUGUST2009 1865
Analysis of Series-Connected Discrete
Josephson Transmission Line
HamidRezaMohebbi,StudentMember,IEEE, and A.HamedMajedi,Member,IEEE
Abstract—Employingageneralizedresistive–capacitiveshunted signal from a dc-bias voltage. Since the inductance associated
junctionmodelforJosephsonjunctions(JJs),thenonlinearwave with each JJ is quite small, an array or stack of JJs can be
propagation in the series-connected discrete Josephson trans-
effectively used to achieve any desired amount of inductance
mission line (DJTL) is investigated. A DJTL consists of a finite
[6], [7]. Incorporation of an unbiased array of JJs in a typical
numberofunitcells,eachincludingasegmentofsuperconducting
transmission line with a single array stack, or generally a block superconducting TL will produce an ultra-low-loss nonlinear
including an identical lumped JJ element. As the governing TL. Propagation characteristics of such nonlinear TLs are
nonlinearwavepropagationisasystemofnonlinearpartialdiffer- highly dependent on the collective behavior of the JJs that
ential equations with mixed boundary conditions, the method of
are in either series connection or parallel connection geome-
thefinitedifferencetimedomainisusedtosolvetheequations.By
tries across the TL. These structures are called either series-
thisnumericaltechnique,thebehaviorofwavepropagationalong
the DJTL can be monitored in time and space domains. Cutoff or parallel-connected discrete Josephson transmission line
propagation, dispersive behavior, and shock-wave formation (DJTL). Although the parallel-connected DJTL has already
throughtheDJTLisaddressedinthispaper. been investigated in the past [7], more emphasis was placed
Index Terms—Dispersion equation, finite-difference time-do- onthestudyofnonlinearfluxondynamicsforrapidsingleflux
main(FDTD)method,Josephsonjunction(JJ)devices,microwave quantum (RSFQ) applications rather than microwave device
superconductivity, nonlinear inductance, nonlinear transmission applications [8]. Moreover, the analysis of such a structure
lines(TLs),nonlinearwavepropagation,shockwaves.
was previously performed based on circuit analysis [7], [9] or
frequency-domaintechniques[10].Inthispaper,weuseanon-
linearfinite-differencetime-domain(FDTD)techniquetosolve
I. INTRODUCTION
TL equations in order to monitor transient and steady-state
S IGNIFICANT improvements on the performance of a responseoftheseries-connectedDJTL.
wide variety of passive microwave devices and systems WeaimtodevelopasystematicstudyofJJ-basedmicrowave/
can be achieved by using superconducting materials due to millimeter-wave and terahertz devices, to take advantage of
their ultra-low surface resistance, frequency-independent pen- theiruniquepropertiesformakingplanarsuperconductivepara-
etration depth, and kinetic inductance. Ultra-low loss, high metric devices and integrated active/passive superconducting
quality factor, and ultra-low dispersive behavior in supercon- microwave/millimeter-wave/terahertz circuits for applications
ducting microwave devices, such as transmission lines (TLs), insuperconductingopto-electronics[11]andquantuminforma-
cavityresonators,bandpassfilters,anddelaylinesarethemain tionprocessing[12]wherehighsensitivityandultra-low-noise
consequencesoftheseproperties.Superconductingmicrowave operationareondemand.
devices have found niche applications in satellite and mobile Inthispaper,wefocusontheanalysisofpropagationcharac-
communication systems, high-quality signal processing sys- teristicsandfeaturesoftheseries-connectedDJTLasthesim-
tems, RADAR [1]–[3], and more recently in circuit cavity plest and the most natural way to incorporate JJs in a typical
quantumelectrodynamicsandquantuminformationprocessors superconducting TL, e.g., microstrip line. In Sections II, the
[4], [5]. circuit model of the JJ and the equivalent nonlinear inductor
Greater flexibility in the design of superconductive pas- are briefly described. In Section III, the physical implemen-
sive and active microwave devices can be obtained by using tation and mathematical analysis of a series-connected DJTL
Josephsonjunctions(JJs).TwobasicelectricalpropertiesofJJs, is presented. Details of the new nonlinear FDTD method to
usefulformicrowavedevices,arenonlinearcurrent-dependent analyze the nonlinear microwave propagation are discussed in
inductivebehaviorandtheabilitytoproduceahigh-frequency Section IV. Section V reports our simulation results based on
the FDTD technique. With this new approach, we observe all
the features associated with a typical nonlinear TL. They in-
ManuscriptreceivedOctober31,2008;revisedApril18,2009.Firstpub-
lishedJuly07,2009;currentversionpublishedAugust12,2009.Thisworkwas cludecutoffpropagation,controllabledispersivebehavior,and
supportedinpartunderanOntarioGraduateScholarship(OGS),byQuantum shock-waveformation.
Works,andbytheInstituteforQuantumComputing(IQC),UniversityofWa-
terloo.
TheauthorsarewiththeDepartmentofElectricalandComputerEngineering II. CIRCUITMODELFORLUMPEDJJ
andtheInstituteforQuantumComputing(IQC),UniversityofWaterloo,Wa-
terloo,ON,CanadaN2L3G1(e-mail:[email protected];ah- AJJisaweaklinkbetweentwosuperconductorelectrodes.
[email protected]). The weak link can be provided by several ways such as a
Colorversionsofoneormoreofthefiguresinthispaperareavailableonline
thin-film insulator, microbridge, or point contact [13]. In the
athttp://ieeexplore.ieee.org.
DigitalObjectIdentifier10.1109/TMTT.2009.2025413 basic JJ, the current that can be driven through the junction is
0018-9480/$26.00©2009IEEE
1866 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.8,AUGUST2009
Fig.2. Series-connectedDJTLonamicrostripline(S–I–Sjunctions).
Fig.1. (a)GeneralizedRCSJmodelofJJ.Theelementdenotedbythecross
signinthismodelisreferredasabasicJJelement.(b)BasicJJisreplacedbya anonlinearoscillatorinalossymedium,asdepictedinFig.(1b),
nonlinearinductance.
whichoscillatesattheplasmafrequency
restrictedtobelessthanthecriticalvalue,whichisdenotedby (8)
. Two canonical relations that describe the circuit model of
a basicJJare Thecharacteristic“qualityfactor”oftheoscillatoris
(1) (9)
(2)
where is called the Stewart–McCumber parameter de-
scribingtheshapeofthedcI–Vcharacteristicsofthejunction.
where is a magnetic flux quanta with a value of
Plasmafrequencyofthejunctiondeterminesthecharacteristic
T m , and is the critical cur-
timescaleofthedynamicalprocessinthejunction.
rent of the JJ. By eliminating the phase difference between
For a 3 m 3 m JJ constructed by Nb–AlO –Nb tech-
twosuperconductors,abasicJJcanbereplacedbyanonlinear
nologyofferedbyHYPRESSinahighcurrentdensityprocess,
inductance [6], as shown in Fig. (1b). This nonlinear inductor
thejunction’sparametersare A, K,
satisfiesthefollowingequations:
K, pF,and [15],[16],whichyields
pH, Trad/s,and .Consid-
(3)
eringanotherjunctionmadeofPb–PbO–Pb,themeasuredjunc-
tion’sparametersare A, K, K,
(4)
fF,and ,thus pH,whichisso
small [13],[17].Therefore,arrays orstacks of JJs are used
toincreasethetotalinductanceofthestructure.Thisarraycan
berepresentedbyasinglejunctionwith timeslargerinduc-
where
tance, times larger resistance, and times smaller capaci-
(5) tancecomparedtoasinglejunction,upontheconditionofiden-
tical junctions, because there exists three distinct channels for
currentflowinatypicalJJ:inductivechannelforcooperpairs,
The advantage of describing the Cooper pairs flow by a resistive channel for normal electron, and capacitive channel
nonlinear inductive channel over the conventional relation fordisplacementcurrent.Thetotalinductivechannels,Cooper
is the fact that we can only deal with voltage pair’s flow, can be delineated by the total phase difference
and current rather than phase difference. The complete elec- acrossthearrayintheform[6]
trical characteristics of the generalized JJ are captured by the
resistive–capacitive shunted junction (RCSJ) circuit model, as (10)
illustrated in Fig. (1a). The equations describing the behavior
The plasma frequency of junctions fabricated by
ofthegeneralizedJJare[14]
Al–Al O –Al technology reduces to the order of
20–100Grad/s;moreover,itpossessesmuchlargerinductance
(6)
and smaller , which is usually referred to as a overdamped
junction with large dissipation (small ) and small
(7)
capacitance [18]. These features are suitable for microwave
superconductingelectronics.
Josephsoncriticalcurrent isafigureofmeritforthejunc-
tion,whichdependsonthequalityofsuperconductorsandthe
III. DJTL
geometryofthejunctions.Accordingto(4),nonlinearityplays
a significant role when the driving current is very close to the To construct a series-connected DJTL, a microstrip line is
critical current; as a result, the effect of nonlinearity becomes loaded in a periodic fashion bya seriesof unbiased JJ blocks,
strongerwhenthecriticalcurrentofthejunctionissmall(typ- assketchedinFig.2.
ically lessthan 10 A). Replacingthe basicJJ element with a Thisblockcanbeasinglejunction, -foldstackedJJs,array
nonlinearinductor,aJosephsontunneljunctioncanbeviewedas oftrilayerjunctions,oranyothercombinationofjunctions.The
MOHEBBIANDMAJEDI:ANALYSISOFSERIES-CONNECTEDDJTL 1867
Fig.3. JJblockwithRCSJmodelofeachjunctioncanberepresentedbya
singleeffectivejunction.
Fig.5. Dispersiondiagramofseries-connectedDJTL.(cid:0) (cid:0)(cid:2)(cid:2)(cid:3)(cid:4)nH,(cid:3) (cid:0)
(cid:3)(cid:2)fF,(cid:4) (cid:0) (cid:5)(cid:2)(cid:6),(cid:5) (cid:0) (cid:3)(cid:2)(cid:2)(cid:2),(cid:4) (cid:0) (cid:5)(cid:2)(cid:6),(cid:3) (cid:0) (cid:7)(cid:7)pF,(cid:6) (cid:0) (cid:5)(cid:2)(cid:6),
(cid:0)(cid:0)(cid:3)(cid:8)(cid:8)nH/m,(cid:3) (cid:0)(cid:8)(cid:8)pF/m,(cid:7)(cid:0)(cid:3)cm.
Fig.4. Unitcellofperiodicallyloadedseries-connectedDJTL. that ,wecanlinearizetheaboveequationsbyletting
.Wetheninsertharmonicsolutions
givenby , forallvariablesof ,
proposedJJblock,whichisusedinoursimulationpart,asde-
, ,and into(11)–(14).Thisprocedureyieldsahomoge-
pictedinFig.3.Itconsistsofanarrayofanidenticalunbiased
nous matrix equation in terms of complex coefficients of ,
junction in parallel to a fit capacitor and also fit shunt re-
, ,and .Thedeterminantofthismatrixshouldvanishin
sistance . These extra fit elements are used to control the
ordertohaveanontrivialsolution.Finally,itresultsinadisper-
resistance, capacitance, and plasma frequency associated with
sionrelationbetweencomplexpropagationconstant
the junction.
andangularfrequency givenby
Thecriticalcurrents,capacitances,normal-statejunctionre-
sistances, and self-inductances are taken tobe identical for all
junctions. Moreover, like an array of JJ, this JJ block can be
representedbyasingleeffectivejunction.TheTLmodelofthis
structureincludingitsunitcellisillustratedinFig.4.
Iftheperiodofthestructure ismuchlessthanthewave- (15)
length of the microwave signal, i.e., , we can Puttingavoltagesource withthe associatedseriesre-
exploit the long wave approximation to form sistance andaloadimpedance attheendsoftheDJTL
a set of differential equations to elucidate the nonlinear mi- andsettingallvariablestozerobefore ,asetofcomplete
crowavepropagationthroughthisstructure.Therefore,inalow- well-posed equations including a system of partial differential
frequencylimit,thisstructurecanbedescribedbyasystemof equations (11)–(14) with mixed boundary conditions and zero
partialdifferentialequationsintheformof initialvaluesintheformof
(11) (16)
(17)
(12)
(18)
Byusing anarrayof1000Al–Al O –Al junctionswithpa-
rameters[18],[19] A, fF, ,
(13)
nH,andfitelementsof and pF,the
(14) dispersiondiagramisshowninFig.5.TheseJosephsonblocks
aremountedona microstriplinewithdistributed
Note that and are lumped elements, but and inductance nH/m and distributed capacitance
are distributed elements. This is the reason of appearance pF/mattheequal-distancepositionswithaspatialperiodof
in (12). is also the flux associated to the nonlinear cm.Thewavelengthat Grad/sisequalto20cm,
inductor (JJ), , , and sotheperiodof cmissmallenoughtoholdtheslow-
. varying approximation, which has been assumed to derive the
Theseequationsarederivedinasimilarmanner,whichisusu- TL(11)–(14).Withtheseparameters,the plasmafrequencyof
ally used to form state equations in circuit theory. Supposing the Josephson block is Grad/s, and the Stewart–
1868 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.8,AUGUST2009
McCumberparameterofasinglejunctionis ,allsuit- where , isthevalueofmatrix evaluatedat
ableformicrowaveapplications.AstheJJblockismodeledby and and isthe4-by-4Jacobianmatrixwhose
aresonantcircuit,theresonancebehaviorisexpectedatplasma entriesaredefinedby
frequency.Atthe low-frequencydomain, theinductivepartof
theJJblockbehavesasashortcircuitandathighfrequenciesthe (22)
capacitivepartoftheblockexhibitsthesamebehavior.There-
fore,inbothregimes,theeffectoftheresistivepartisreduced where and arethe thand thelementincolumnvectors
andweexpectlowattenuation.Ontheotherhand,atthereso- and .Toavoidmidpointevaluations,theJacobianmatrices
nantfrequencyoccurringattheplasmafrequency,inductorand canbefoundby
capacitorcomponentsofeachblockcanceleachother,andthere-
sistancepartbecomesmoreprominentbyinducinglargeattenua- (23)
tion.Furthermore,accordingtoFig.5,weobservenondispersive
behaviorbelowtheplasmafrequency(lowfrequency)andalso (24)
faraboveit(highfrequency).Atlowfrequencies,theinductorel-
ementsaredominantcomponents;however,athighfrequencies, Obviously, by applying an update equation of (21) into the
thecapacitorsofeachblockbecomedominantelements.Thus, endingpointsatthetwoboundaries,twofictitiouspointsappear
slowwavepropagationisexpectedatlowfrequenciesincompar- ateachtimestep.Duetothepossibilityofgeneratinginstability,
isontohighfrequencies.Alltheaboveexpectationsareobserved careshouldbetakentocomputesuchpoints.Thus,weusethe
clearlyinthedispersiondiagramofFig.5. followingrelationstocalculateextra-leftandextra-rightpoints,
respectively,[20]:
IV. FDTDMETHOD
The first step in obtaining an FDTD solution is to set up a (25)
regulargridinspaceandtime.Timeandspacestepsaredenoted (26)
by and ,respectively,andthetotalnumberoftemporaland
spatialgridsinthecomputationaldomainisreferredby and Basedon(16)and(17),updateequationsforboundarycon-
. A few extra points beyond the computation domain might ditionsatthetwoendsoftheTLareasfollows:
beaddedfornumericalreasons.Thenextstepistoapproximate
thedifferentialequationswithaproperfinite-differencescheme. (27)
WeusedanexplicitLax–Wendroffscheme[20],[21],whichis
well suited for our problem. This scheme provides a second- (28)
order accuracy by itself so there is no need to complicate the
implementationbydefiningadditionalgridspointsathalf-time Itcanbeperceivedthat(21)involvesfourunknownsthatare
and half-space [22]–[24]. To apply the Lax–Wendroff scheme coupled to each other through four nonlinear equations so at
inourmodel,(11)–(14)arerestatedinthematrixform each grid in the computational domain, a system of nonlinear
simultaneousequationsmustbesolved.Adetailedprocedureof
(19) theFDTDimplementationisillustratedintheflowchartshown
inFig.6.
wherecolumnvectors , ,and aredefinedas
, and V. NUMERICALRESULTS
Formanynumericalsimulations,whenverysmallorverybig
numbersareinvolved,itisoftenhelpfultonormalizeallparam-
etersandvariablestospecialvalues.ThescalingrulesforFDTD
analysis of the DJTL is described in Table I. Basically, any
scaling rule must have this important property such that when
(20) wesubstitutenewnormalizedvariablesandparametersintothe
Note that is a nonlinear function of . Applying the setofmasterequationsfortheproblem,theseequationsholdthe
Lax–Wendroffscheme,theupdateequationcanbeobtainedas sameformastheyhaveforthenonnormalizedvariablesandpa-
follows: rameters.Hence,inordertoestablishanormalizationruleinour
problem,wechoosefourarbitraryconstants,namely, , , ,
and , to normalize frequency, wavenumber, impedance, and
current by dividing them by these constants, respectively. All
other remaining parameters and variables are then normalized
into the proper form by using these four assumed parameters,
as described in Table I. By putting new normalized variables
into(11)–(14)orthedispersionrelationof(15),thisconclusion
isdrawnfromtheabovediscussionthat , ,and canbe
takenasarbitraryconstants,but mustbeequalsto1m .A
(21) summaryoftheaboveprocessisshowninTableI.
MOHEBBIANDMAJEDI:ANALYSISOFSERIES-CONNECTEDDJTL 1869
Fig.7. WavepropagationinaDJTLanalyzedbytheRCSJmodel,(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:2)
(cid:3)m ,(cid:3)(cid:0) (cid:2)(cid:3)(cid:0) (cid:2)(cid:3)(cid:0) (cid:2)(cid:3),(cid:2)(cid:0) (cid:2)(cid:4)(cid:4)(cid:5),(cid:0)(cid:0) (cid:2)(cid:3),(cid:5)(cid:0) (cid:2)(cid:6),(cid:6)(cid:0) (cid:2)(cid:4)(cid:4)(cid:3),
(cid:7)(cid:0) (cid:2)(cid:7),(cid:8)(cid:0) (cid:2)(cid:7),(cid:9)(cid:2)(cid:4)(cid:4)(cid:4)(cid:3),(cid:10) (cid:2)(cid:4)(cid:4)(cid:4)(cid:4)(cid:3).
Fig.6. FlowchartincludingalldetailsforexplicitimplementationofFDTDto
analyzetheDJTL.
TABLEI
NORMALIZATIONRULE
Fig.8. GroupandphasevelocityforwavepropagationinaDJTLbasedonthe
RCSJmodel.
, m , , and
.
Fig.7illustratesthevoltagewavepropagationinaseries-con-
nectedDJTLoverbothspaceandtimeaxes.Duetotheabrupt
jump from the resting initial condition to some values by the
voltage source, many Fourier components (frequency compo-
In order to conduct the FDTD simulation, we choose the nents)areexcited;hence,weobservedispersivebehaviorinthe
samestructureandalsothesamephysicalandgeometricalpa- forefrontofthewaveasmoreclearlyshowninFig.8,whichis
rameters as described in Section III for calculating the disper- thetopperspectiveofFig.7.
siondiagramofFig.5.ThenormalizationruleofTableIisap- Duetothenormalresistivechannel ,thewavewillatten-
plied into the actual parameters and variables of the problem uategraduallyassketchedinthevoltageprofileofFig.9.How-
by reference parameters , rad/s, ever,theleadingcycleofthewavetraindecaysmorecompared
A, and m . To be in a small am- toothercyclesbecauseofthedispersioneffectthatbroadensit.
plitude regime, we drive the structure by a sinusoidal voltage Bymeasuringthedistancebetweentwosuccessivecrestsofthe
source with amplitude of V and frequency of wavedepictedinFig.9,thephaseconstantofthewaveisfound
Grad/s( GHz).Theseriesresistanceasso- tobe rad/m.Moreover,bysimplealgebraiccalcula-
ciatedwiththevoltagesourceandalsotheloadimpedanceatthe tionbasedonthedataofFig.9,theattenuationconstantisgiven
endofthestructureare .Afternormal- as Np/m.Both and areinagreementwiththere-
ization, the new variables are given as , , sultshownindispersiondiagramofFig.5.Themagnitudeofthe
1870 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.57,NO.8,AUGUST2009
Fig.9. Profileofthevoltagepatterninseries-connectedDJTL.Attenuationand
Fig.11. Whendrivingfrequencyiscloseenoughtoplasmafrequency,DJTL
phaseconstantcanbefoundfromthisfigure.
revealscutoffpropagation.
OnekindofcutoffconditionhappenswhentheDJTLisdriven
with frequencies very close to the plasma frequency. At these
frequencies, resonance occurs in the series JJ blocks and the
line becomes very lossy; therefore, the wave decays very fast.
Fig.11showsthecutoffpropagationwhenthefrequencyofthe
voltagesourceis Grad/s.Thisfrequencyislocatedin
theintervalofthedispersion curve(Fig.5), whereattenuation
is large.
Spatialdiscretenesscancauseanother typeofcutoff,which
happens at the Brag frequency. This observation is similar to
thecutoffpropagationintheparallelDJTL,whichhasbeende-
scribed by the discreteness factor [7], [25], [26]. This fact is
fully explained in the Appendix. To simplify the problem, we
replaceallJosephsonblockswithbasicjunctionsbyassuming
Fig.10. WavepacketpropagationinDJTL,(cid:0)(cid:0) (cid:2)(cid:3)(cid:4),(cid:0)(cid:0) (cid:2)(cid:4). that , andremovingallfitelements ,
.Inthiscase,thepropagationconditionisgivenby
voltageatthebeginningofthelineishalfofthemagnitudeof (30)
thevoltagesource,asseeninFig.9,becauseoftheimpedance
matchingbetweenthesourceandline.
For example, by setting m , ,
Thestudyofthewavepacketintroducesanotherinteresting
andsinusoidal sourcewith frequency , these
aspect of the DJTL. In the generalized RCSJ model of the JJ,
parameters fail to satisfy (30) and instead of propagation we
theresistiveelementcausesthedispersionbehavior,whichhas
haveacutoffpropagationinthesteady-statesolutionoftheanal-
alreadybeenseeninFigs.7and8.Thisdispersivebehaviorcan
ysis,asillustratedinFig.12for .Accordingto(30),
bemonitoredbythewavepacket.Thewavepacketthatweuse
atagivenfrequency,byincreasing , ,or ,thecutofffre-
isinthe formof
quencydecreases,soforlargevaluesofcircuitparameters,we
(29) encounter blocking in wave propagation at lower frequencies.
This fact has been reported for the parallel-connected DJTL
where . The wave is a relatively smooth [25],[26].
function and plays the role of an envelope for the wave func- According to (4), nonlinear Josephson inductance increases
tion . The envelope travels at the group velocity and withincreasingcurrent,soweexpectthathigh-currentsections
the crestsof the wavefunction moveswith phasevelocity. As of the waveform to propagate slower than the low-current
observedinFig.10,atdifferentzero-crossingpointsoftheen- sections. Qualitatively, as time evolves, the peak of a current
velope,thephaseofthewavefunctionchanges,andthisisevi- (orvoltagesincebothhavethesameprofile)leavesbehindthe
denceofdispersivebehavior. bottom. As a result, a wave with a steeping end can develop,
Numericalsimulationsrevealthisimportantfactthat,insome which eventually leads to a jump discontinuity [27], as repre-
particularcases,thecutoffconditionhappensintheseriesDJTL. sentedinFig.13.Thistypeofwave,whichtakestheformofa
MOHEBBIANDMAJEDI:ANALYSISOFSERIES-CONNECTEDDJTL 1871
Fig.14. DiscretecircuitmodelofDJTL.
theDJTLhasdemonstratedmoremicrowavecompatibilitybe-
causeofitsimplementationonaregularTL.AdiscreteJJblock
canbeanycombinationofJJsandcircuitelements,aspointed
outinthispaper.Moreover,asampleofaJJblockwithitsprac-
ticalparameterswasillustrated.Dispersionequationshavebeen
derivedanddifferentregimesbasedonthedispersiondiagram
andplasmafrequencyhavebeendiscussed.Arigorous,robust,
andstablenonlinearFDTDbasedontheexplicitLax–Wendroff
Fig.12. Stopped-propagationofvoltagewavethroughaDJTL,(cid:0)(cid:0) (cid:2) (cid:2)(cid:0) (cid:2) scheme has been developedto solve the nonlinear waveequa-
(cid:3)m ,(cid:0)(cid:0) (cid:2)(cid:3)(cid:4)m ,(cid:3)(cid:2)(cid:4)(cid:4)(cid:4)(cid:5),(cid:5)(cid:2)(cid:4)(cid:4)(cid:4)(cid:4)(cid:5)(cid:6)(cid:0) (cid:2)(cid:6),(cid:7)(cid:0) (cid:2)(cid:4)(cid:4)(cid:5). tions.Goodagreementbetweentheresultsofthedispersiondia-
gram,whichisbasedontheanalyticaltreatmentofthestructure
inthefrequencydomainandtheresultsoftheFDTDsolverin
timedomainhasbeendemonstrated.Thecutoffpropagationdue
totheresonancebehavioroftheJosephsonblockandalsodueto
thediscretenessofthestructurehasbeendescribed.Shock-wave
formationhasbeenobservedwhentheDJTLwasexcitedsuch
thattheflowingcurrentisveryclosetothecriticalcurrentofthe
junctions.Thisisanindicationoftheexistenceofahighnon-
linearpropertyinatypicalDJTLwithapotentialapplicationin
realizing parametric devices such as traveling-wave amplifiers
and mixers.
APPENDIX
By dividing a series-connected DJTL into identical unit
cells, we can have another view of the DJTL, as illustrated in
Fig.14.Insteadofcontinuousvariable ,index isdesignated
Fig.13. SketchoftheformationofashockwaveinanonlinearJJtransmission
line,(cid:3)(cid:2)(cid:5)(cid:0)(cid:3)(cid:4) ,(cid:7)(cid:8)(cid:9)(cid:10)(cid:2)(cid:4)(cid:4)(cid:4)(cid:4)(cid:11)(cid:12),(cid:0)(cid:0)(cid:2)(cid:2)(cid:0)(cid:2)(cid:3)m ,(cid:0)(cid:0) (cid:2)(cid:3). foreachunitcell.
Similartotheparallel-connectedDJTL[7],weattainthefol-
lowingequationtoexpressfluxpropagationinthestructure:
very sharp change, is called as a shock wave. To see this, the
voltagesourceischosentobeaGaussianpulseintheformof
(31)
Thefullwavehalfmaximum(FWHM)oftheGaussianpulse
hastherelation (33)
where is the flux associated to the JJ in the th segment.
(32)
In above equation, the first, second, third, and fourth deriva-
Wechoose [28],where isanormalized tives of with respect to time are denoted by , , ,
time step, which is 2.82 10 , and some other parameters and , respectively. Considering a particular harmonic so-
are mentioned in Fig. 13. We have reduced the effect of the lution and small amplitude approximation
numerical dispersion, as displayed in Fig. 13, by having fine , substituting this into (33), this yields the fol-
gridding and also running at a rate very close to the stability lowingdispersionrelation:
conditionofCourant–Friedrichs–Lewy(CFL).
VI. CONCLUSION
In this paper, a series-connected DJTL has been analyzed
basedonTLtheory.ComparedtothecontinuousJosephsonTL, (34)