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PRACTICALRFSYSTEM DESIGN CONTENTS
5NOISEANDNONLINEARITY123 5.1IntermodulationofNoise/123
5.1.1Preview/124
5.1.2FlatBandpassNoise/125
5.1.3Second-OrderProducts/125
5.1.4Third-OrderProducts/130
5.2CompositeDistortion/133
5.2.1Second-OrderIMs(CSO)/134
5.2.2Third-OrderIMs(CTB)/136
5.2.3CSOandCTBExample/136
5.3DynamicRange/137
5.3.1Spurious-FreeDynamicRange/137
5.3.2OtherRangeLimitations/139
5.4OptimizingCascades/139
5.4.1CombiningParametersonOneSpreadsheet/139
5.4.2OptimizationExample/143
5.5SpreadsheetEnhancements/146
5.5.1LookupTables/146
5.5.2UsingControls/147
5.6Summary/147 Endnotes/147
6ARCHITECTURESTHATIMPROVELINEARITY149 6.1ParallelCombining/149
6.1.190◦ Hybrid/150
6.1.2180◦ Hybrid/152
6.1.3SimplePush–Pull/154
6.1.4Gain/155
6.1.5NoiseFigure/156
6.1.6CombinerTrees/156
6.1.7CascadeAnalysisofaCombinerTree/157
6.2Feedback/158
6.3Feedforward/159
6.3.1IntermodsandHarmonics/160
6.3.2Bandwidth/161
6.3.3NoiseFigure/161
6.4NonidealPerformance/162
6.5Summary/163 Endnotes/163
7FREQUENCYCONVERSION165 7.1Basics/165
7.1.1TheMixer/165
7.1.2ConversioninReceivers/167
7.1.3Spurs/168
7.1.4ConversioninSynthesizersandExciters/170
7.1.5Calculators/170
7.1.6DesignMethods/170
7.1.7Example/171
7.2SpuriousLevels/171
7.2.1DependenceonSignalStrength/171
7.2.2EstimatingLevels/173
7.2.3StrategyforUsingLevels/175
7.3Two-SignalIMs/176
7.4PowerRangeforPredictableLevels/177
7.5SpurPlot,LOReference/180
7.5.1SpreadsheetPlotDescription/180
7.5.2ExampleofaBandConversion/182
7.5.3OtherInformationonthePlot/184
7.6SpurPlot,IFReference/186
7.7ShapeFactors/196
7.7.1Definitions/197
7.7.2RFFilterRequirements/197
7.7.3IFFilterRequirements/200
7.8DoubleConversion/202
7.9OperatingRegions/203
7.9.1AdvantageousRegions/203
7.9.2LimitationonDownconversion, Two-by-Twos/206
7.9.3HigherValuesof m /209
7.10Examples/211
7.11NoteonSpurPlotsUsedinThisChapter/216
7.12Summary/216
Endnotes/217
8CONTAMINATINGSIGNALSINSEVERENONLINEARITIES219 8.1Decomposition/220
8.2HardLimiting/223
8.3SoftLimiting/223
8.4Mixers,ThroughtheLOPort/225
8.4.1AMSuppression/225
8.4.2FMTransfer/226
8.4.3Single-SidebandTransfer/226
8.4.4MixingBetweenLOComponents/228
8.4.5TroublesomeFrequencyRangesintheLO/228
8.4.6SummaryofRanges/235
8.4.7EffectonNoiseFigure/236
8.5FrequencyDividers/240
8.5.1SidebandReduction/240
8.5.2Sampling/241
8.5.3InternalNoise/242
8.6FrequencyMultipliers/242
8.7Summary/243 Endnotes/244
9PHASENOISE245 9.1DescribingPhaseNoise/245
9.2AdverseEffectsofPhaseNoise/247
9.2.1DataErrors/247
9.2.2Jitter/248
9.2.3ReceiverDesensitization/249
9.3SourcesofPhaseNoise/250
9.3.1OscillatorPhaseNoiseSpectrums/250
9.3.2IntegrationLimits/252
9.3.3RelationshipBetweenOscillator Sϕ and Lϕ /252
9.4ProcessingPhaseNoiseinaCascade/252
9.4.1FilteringbyPhase-LockedLoops/253
9.4.2FilteringbyOrdinaryFilters/254
9.4.3ImplicationofNoiseFigure/255
9.4.4TransferfromLocalOscillators/255
9.4.5TransferfromDataClocks/256
9.4.6IntegrationofPhaseNoise/258
9.5DeterminingtheEffectonData/258
9.5.1ErrorProbability/258
9.5.2ComputingPhaseVariance,Limitsof Integration/259
9.5.3EffectoftheCarrier-RecoveryLooponPhase Noise/260
9.5.4EffectoftheLooponAdditive Noise/262
9.5.5ContributionofPhaseNoisetoData Errors/263
9.5.6EffectsoftheLow-FrequencyPhase Noise/268
9.6OtherMeasuresofPhaseNoise/269
9.6.1Jitter/269
9.6.2AllanVariance/271
9.7Summary/271
Endnote/272
APPENDIXAOPAMPNOISEFACTORCALCULATIONS273
A.1InvarianceWhenInputResistorIsRedistributed/273
A.2EffectofChangeinSourceResistances/274
A.3Model/276
APPENDIXBREPRESENTATIONSOFFREQUENCYBANDS, IFNORMALIZATION279
B.1Passbands/279
B.2AcceptanceBands/279
B.3FilterAsymmetry/286
APPENDIXCCONVERSIONARITHMETIC289
C.1ReceiverCalculator/289
C.2SynthesisCalculator/291
APPENDIXEEXAMPLEOFFREQUENCYCONVERSION293
APPENDIXFSOMERELEVANTFORMULAS303
F.1Decibels/303
F.2ReflectionCoefficientandSWR/304
F.3CombiningSWRs/306
F.3.1SummaryofResults/306
F.3.2Development/307
F.3.3MaximumSWR/308
F.3.4MinimumSWR/309
F.3.5RelaxingRestrictions/309
F.4ImpedanceTransformationsinCables/310
F.5SmithChart/310
APPENDIXGTYPESOFPOWERGAIN313
G.1AvailableGain/313
G.2MaximumAvailableGain/313
G.3TransducerGain/314
G.4InsertionGain/315
G.5ActualGain/315
APPENDIXHFORMULASRELATINGTOIMsAND HARMONICS317
H.1SecondHarmonics/317
H.2Second-OrderIMs/318
H.3ThirdHarmonics/318
H.4Third-OrderIMs/319
H.5DefinitionsofTerms/320
APPENDIXICHANGINGTHESTANDARDIMPEDANCE321
I.1GeneralCase/321
I.2UnilateralModule/323
APPENDIXLPOWERDELIVEREDTOTHELOAD325
APPENDIXMMATRIXMULTIPLICATION327
APPENDIXNNOISEFACTORS—STANDARDAND THEORETICAL329
N.1TheoreticalNoiseFactor/329
N.2StandardNoiseFactor/331
N.3StandardModulesandStandardNoiseFactor/332
N.4ModuleNoiseFactorinaStandardCascade/333
N.5HowCanThisBe?/334
N.6NoiseFactorofanInterconnect/334
N.6.1NoiseFactorwithMismatch/335
N.6.2InMoreUsableTerms/336
N.6.3Verification/338
N.6.4ComparisonwithTheoreticalValue/340
N.7EffectofSourceImpedance/341
N.8RatioofPowerGains/342
Endnote/343
APPENDIXPIMPRODUCTSINMIXERS345
APPENDIXSCOMPOSITE
APPENDIXTTHIRD-ORDERTERMSATINPUTFREQUENCY353
APPENDIXVSENSITIVITIESANDVARIANCEOFNOISE
APPENDIXZNONSTANDARDMODULES363
Z.1GainofCascadeofModulesRelativetoTestedGain/363
Z.2FindingMaximumAvailableGainofaModule/366
Z.3Interconnects/367
Z.4Equivalent S Parameters/367
Z.5 S ParametersforCascadeofNonstandardModules/368 Endnote/369
REFERENCES371
Endnote/377
PREFACE ThisbookisaboutRFsystemanalysisanddesignatthelevelthatrequiresan understandingoftheinteractionbetweenthemodulesofasystemsotheultimate performancecanbepredicted.Itdescribesconceptsthatareadvanced,thatis, beyondthosethataremorecommonlytaught,becausethesearenecessarytothe understandingofeffectsencounteredinpractice.Itisaboutansweringquestions suchas:
• Howwillthegainofacascade(agroupofmodulesinseries)beaffected bythestanding-waveratio(SWR)specificationsofitsmodules?
• Howwillnoiseonalocaloscillatoraffectreceivernoisefigureanddesensitization?
• Howdoestheeffectivenoisefigureofamixerdependonthefilteringthat precedesit?
• Howcanwedeterminethelinearityofacascadefromspecificationson itsmodules?
• Howdoweexpectintermodulationproducts(IMs)tochangewithsignal amplitudeandwhydotheysometimeschangedifferently?
• Howcanmodulesbecombinedtoreducecertainintermodulationproducts ortoturnbadimpedancematchesintogoodmatches?
• Howcanthespuriousresponsesinaconversionschemebevisualizedand howcanthemagnitudesofthespursbedetermined?Howcanthispicture beusedtoascertainfilterrequirements?
• Howdoesphasenoiseaffectsystemperformance;whatareitssourcesand howcantheeffectsbepredicted?
IwillexplainmethodslearnedovermanyyearsofRFmoduleandsystemdesign, withemphasisonthosethatdonotseemtobewellunderstood.Someareavailableintheliterature,somewerepublishedinreviewedjournals,somehave developedwithlittleexposuretopeerreview,butallhavebeenfoundtobe importantinsomeaspectofRFsystemengineering.
IwouldliketothankEricUnruhandBillBeardenforreviewingpartsof themanuscript.Ihavealsobenefitedgreatlyfromtheopportunitytoworkwith manyknowledgeablecolleaguesduringmyyearsatSylvania-GTEGovernment SystemsandatESL-TRWintheSantaClara(Silicon)Valleyandwouldlike tothankthem,andthoseexcellentcompaniesforwhichweworked,forthat opportunity.IamalsogratefulfortheeducationthatIreceivedatSantaClara andStanfordUniversities,oftenwiththehelpofthosesamecompanies.However, onlyIbeartheblameforerrorsandimperfectionsinthiswork.
WILLIAM F.EGAN
Cupertino,California February,2003
GETTINGFILESFROMTHEWILEYftp ANDINTERNETSITES Todownloadspreadsheetsthatarethebasesforfiguresinthisbook,useanftp programoraWebbrowser.
FTPACCESS Ifyouareusinganftpprogram,typethefollowingatyourftpprompt:
ftp://ftp.wiley.com
Someprogramsmayprovidethefirst“ftp”foryou,inwhichcasetype
ftp.wiley.com
Loginasanonymous(e.g.,UserID:anonymous).Leavepasswordblank.After youhaveconnectedtotheWileyftpsite,navigatethroughthedirectorypathof: /public/sci_tech_med/rf_system
WEBACCESS IfyouareusingastandardWebbrowser,typeURLaddressof: xix
xx GETTINGFILESFROMTHEWILEYftpANDINTERNETSITES
ftp://ftp.wiley.com
Navigatethroughthedirectorypathof: /public/sci_tech_med/rf_system
Ifyouneedfurtherinformationaboutdownloadingthefiles,youcancallWiley’s technicalsupportat201-748-6753.
SYMBOLSLISTANDGLOSSARY Thefollowingisalistoftermsandsymbolsusedthroughoutthebook.Special meaningsthathavebeenassignedtothesymbolsaregiven,althoughthesame symbolssometimeshaveothermeanings,whichshouldbeapparentfromthe contextoftheirusage.(Forexample, A and B canbeusedforamplitudesofsine waves,inadditiontothespecialmeaningsgivenbelow.)
≡ isidenticallyequalto,ratherthanbeingequalonlyunder someparticularcondition
= isdefinedas
∼ (superscript)indicatesrms
X |y variable X withthecondition y orreferringto y
X |y 2 y 1 variable X with y between y land y 2
x angleorphaseof x
∼ low-passfilter
∼ band-passfilter
acceptancebandbandoffrequenciesbeyondthepassbandwhererejection isnotrequired;usedtoindicatetheregionbetween thepassbandandarejectionband contaminantundesiredRFpower passbandbandoffrequenciesthatpassthroughafilterwith minimalattenuationorwithlessthanaspecified attenuation
rejectionbandbandoffrequenciesthatarerejectedorreceivea specifiedattenuation(rejection) sidebandsignalinrelationtoalargersignal
GenericSymbols(appliedtoothersymbols)
*complexconjugate
|x | magnitudeorabsolutevalueof x
˘ xx isanequivalentnoisefactororgainthatcanbeusedinstandard equationstorepresentcascadeswithextrememismatches(see Section3.10.4)
ParticularSymbols A voltagegainindB.Notethat G canaswellbeusedif impedancesarethesameorthevoltageisnormalizedto R0 . a voltagetransferratio.
|a | voltagegain(notindB)
AMamplitudemodulation
an nth-ordertransfercoefficient[seeEq.(4.1)]
aRT round-tripvoltagetransferratio
B noisebandwidth
Br RFbandwidth
Bv video,orpostdetection,bandwidth
BWbandwidth
c(n,j ) j thbinomialcoefficientfor (a + b)n (Abromowitzand Stegun,1964,p.10)
cassubscriptreferringtocascade
CATVcabletelevision
cblsubscriptreferringtocable
CSOcompositesecond-orderdistortion(Section5.2)
CTBcompositetriple-beatdistortion(Section5.2)
dBdecibels
DBMdoublybalancedmixer
dBmdecibelsreferencedto1mW
dBcdecibelsreferencedtocarrier
dBVdecibelsreferencedto1V
dBWdecibelsreferencedto1W
e voltagefromaninternalgenerator F noisefigure, F = 10dBlog10 f orfundamental(asopposed toharmonicorIM).
f noisefactor(notindB)orstandardnoisefactor(measured withstandardimpedances)orfrequency
ˆ f theoreticalnoisefactor(measuredwithspecifieddriving impedance)(seeSections3.1,N.1)
FDMfrequencydivisionmultiplex
fc centerfrequency
fosc oscillatorcenterfrequency
fI or fIF intermediatefrequency, frequencyatamixer’soutput
fL or fLO localoscillatorfrequency
FMfrequencymodulation
fm modulationfrequency
fR or fRF radiofrequency,thefrequencyatamixer’sinput
G powergain,sometimesgainingeneral,indB.
gk powergainofmodule k ,sometimesgainingeneral,notindB.
gpk powergainprecedingmodule k
H subscriptreferringtoharmonic
I ,IFintermediatefrequency,theresultofconvertingRFusinga localoscillator
i subscriptindicatingasignaltravelinginthedirectionofthe systeminput
IFintermediatefrequency,frequencyatamixer’soutput
IIPinputinterceptpoint(IPreferredtoinputlevels)
IMintermodulationproduct(intermod)
IMnnth-orderintermodorIMformodule n insubscriptindicatingasignalenteringamodule(1)attheport ofconcernor(2)attheinputport int(x) integerpartof x
IPinterceptpoint
IPn interceptpointfor nth-ordernonlinearityorformodule n
ISFDRinstantaneousspur-freedynamicrange(seeSection5.3)
k Boltzmann’sconstant
kT0 approximately4 × 10 21 W/Hz
L single-sidebandrelativepowerdensity
L,LOlocaloscillator,thegenerallyrelativelyhigh-powered, controllable,frequencyinafrequencyconversionorthe oscillatorthatprovidesit
Lϕ single-sidebandrelative powerdensityduetophasenoise
M amatrix(boldformatindicatesavectorormatrix)
m modulationindex(seeSection8.1)
˜ m rmsphasedeviationinradians masubscriptfor“maximumavailable”
MAX{a,b } thelargerof a or b
m × nm referstotheexponentoftheLOvoltageand n referstothe exponentoftheRFvoltageintheexpressionforaspurious product;ifwritten,forexample,3 × 4, m is3and n is4
N0 noisepowerspectraldensity
NT availablethermalnoisepowerspectraldensityat290K, kT0
o subscriptindicatingasignaltravelinginthedirectionofthe systemoutput.
OIPoutputinterceptpoint(IPreferredtooutputlevels)
outsubscriptindicatingasignalexitingamodule(1)attheport ofconcernor(2)attheoutputport
P powerindB.
p power(notindB).
pavail,j availablepoweratinterface j (precedingmodule j )
PMphasemodulation
pout,j outputpoweratinterface j (precedingmodule j )
PPSDphasepowerspectraldensity
PSDpowerspectraldensity
R ,RFradiofrequency,thefrequencyatamixer’sinput
R0 agreed-uponinterfaceimpedance,astandardimpedance(e.g., 50 );characteristicimpedanceofatransmissionline
RTsubscriptfor“roundtrip”
S powerspectraldensityor S parameter(seeSection2.2.1)
ˆ S sensitivity(seeSection2.5)
Sijk
S parameterofrow i andcolumn j intheparametermatrix formodule(orelement)number k
SFshapefactor,ratioofbandwidthwhereanattenuationis specifiedtopassbandwidth
SFDRspur-freedynamicrange(seeSection5.3.1)
S/N signal-to-noisepowerratio
SSBsingle-sideband;referstoasinglesignalinrelationtoalarger signal
SWRstandingwaveratio(seeSectionF.2)
T absolutetemperatureorsubscriptreferringtoconditions duringtest
T0 temperatureof290K(16.85◦ C)
Tijk
T parameter(seeSection2.2.3)ofrow i andcolumn j inthe parametermatrixformodule(orelement)number k
Tk noisetemperatureofmodule k (seeSection3.2)
UUTunitundertest
V avector(boldformatindicatesavectorormatrix)
v normalizedwavevoltage(seeSection2.2.2)orvoltage(notin dB.)
V voltageindB
ˆ
v phasorrepresentingthewavevoltage(seeSection2.2.2)
v phasorwhosemagnitudeisthermsvalueofthevoltage v =ˆv/√2(seeSection2.2.2)
vi , vin , vo , vout seeFig.2.2andSection2.2.1 ± maximum ± deviationindBofcablegain Acbl ,fromthemean f peakfrequencydeviationorfrequencyoffsetfromspectral center
ρ reflectioncoefficient(seeSectionF.2)
σ standarddeviation
σ 2 variance
τ voltagetransferratioofamatchedcable(i.e.,noreflectionsat theends)
ϕ(t)ωt + θ
Practical RF System Design. William F. Egan Copyright 2003 John Wiley & Sons, Inc.
0-471-20023-9
CHAPTER1 INTRODUCTION Thisbookisaboutsystemsthatoperate atradiofrequencies(RF)(including microwaves)wherehigh-frequencytechniques,suchasimpedancematching,are important.ItcoverstheinteractionsoftheRFmodulesbetweentheantenna outputandthesignalprocessors.Itsgoalistoprovideanunderstandingofhow theircharacteristicscombinetodeterminesystemperformance.Thischapterisa generaldiscussionoftopicsinthebookandofthesystemdesignprocess.
1.1SYSTEMDESIGNPROCESS Wedosystemdesignbyconceptualizingasetoffunctionalblocks,andtheir specifications,thatwillinteractinamannerthatproducestherequiredsystem performance.Todothissuccessfully,werequireimaginationandanunderstandingofthecostsofachievingthevariousspecifications.Ofcourse,wealsomust understandhowthecharacteristicsoftheindividualblocksaffecttheperformance ofthesystem.Thisisessentiallyanalysis,analysisattheblocklevel.Bythis process,wecancombineexistingblockswithnewblocks,usingthespecificationsoftheformerandcreatingspecificationsforthelatterinamannerthatwill achievethesystemrequirements.
Thespecificationsforablockgenerallyconsistoftheparametervalueswe wouldlikeittohaveplusallowedvariations,thatis,tolerances.Wewouldlike thetolerancestobezero,butthatisnotfeasiblesoweacceptvaluesthatare compromisesbetweencostsandresultingdegradationsinsystemperformance. Notuntilmoduleshavebeendevelopedandmeasureddoweknowtheirparameterstoahighdegreeofaccuracy(atleastforonecopy).Atthatpointwemight insertthemoduleparametersintoasophisticatedsimulationprogramtocompute
CHAPTER1INTRODUCTION theexpectedcascadeperformance(orperhapsjusthookthemtogethertosee howthecascadeworks).Butitisimportantinthedesignprocesstoascertain therangeofperformancetobeexpectedfromthecascade,givenitsmodule specifications.Weneedthisabilitysowecanwritethespecifications.
Spreadsheetsareusedextensivelyinthisbookbecausetheycanbehelpfulin improvingourunderstanding,whichisourmainobjective,whilealsoproviding toolstoaidintheapplicationofthatunderstanding.
1.2ORGANIZATIONOFTHEBOOK ItiscommonpracticetolistthemodulesofanRFsystemonaspreadsheet, alongwiththeirgains,noisefigures,andinterceptpoints,andtodesigninto thatspreadsheetthecapabilityofcomputingparametersofthecascadefrom thesemoduleparameters.Thespreadsheetthenservesasaplanforthesystem. Thenextthreechaptersaredevotedtothatprocess,onechapterforeachof theseparameter.
Atfirstitmayseemthatoverallgaincanbeeasilycomputedfromindividual gains,buttheusualimperfectimpedancematchescomplicatetheprocess.In Chapter2,wediscoverhowtoaccountfortheseimperfections,eitherexactly or,inmostcases,byfindingtherangeofsystemgainsthatwillresultfromthe rangeofmoduleparameterspermittedbytheirspecifications.
Themethodforcomputingsystemnoisefigurefrommodulenoisefigures iswellknowntomanyRFengineersbutsomesubtletiesarenot.Ideally,we usenoisefigurevaluesthatwereobtainedunderthesameinterfaceconditions asseeninthesystem.Practically,thatinformationisnotgenerallyavailable, especiallyatthedesignconceptphase.InChapter3,weconsiderhowtousethe informationthatisavailabletodeterminesystemnoisefigureandwhatvariations aretobeexpected.Wealsoconsiderhowtheeffectivenoisefiguresofmixers areincreasedbyimagenoise.Laterwewillstudyhowthelocaloscillator(LO) cancontributetothemixer’snoisefigure.
Theconceptofinterceptpoints,howtouseinterceptpointstocomputeintermodulationproducts,andhowtoobtaincascadeinterceptpointsfromthoseofthe moduleswillbestudiedinChapter4.Anomalousintermodsthatdonotfollow theusualrulesarealsodescribed.
Thecombinedeffectsofnoiseandintermodulationproductsareconsidered inChapter5.Oneresultistheconceptofspur-freedynamicrange.Anotheris theportrayalofnoisedistributionsresultingfromtheintermodulationofbands ofnoise.Thesimilaritybetweennoise bandsandbandsofsignalsbothaidsthe analysisandprovidespracticalapplicationsforit.
Havingestablishedthemeansforcomputingparametersforcascadesofmodulesconnectedinseries,inChapter6wetakeabriefjourneythroughvariousmeansofconnectingmodulesorcomponentsinparallel.Wediscoverthe advantagesthatthesevariousmethodsprovideinsuppressingspuriousoutputs andhowtheiroverallparametersarerelatedtotheparametersoftheindividualcomponents.
Then,inChapter7,weconsiderthemethodfordesignoffrequencyconverters thatusesgraphstogiveanimmediatepictureofthespursandtheirrelationships tothedesiredsignalbands,allowingustovisualizeproblemsandsolutions.We alsolearnhowtopredictspuriouslevels andthose,alongwiththerelationships betweenthespursandthepassbands,permitustoascertainfilterrequirements.
Theprocessesdescribedintheinitialchaptersarelinear,oralmostso,except forthefrequencytranslationinherentin frequencyconversion.Someprocesses, however,areseverelynonlinearand,whileperformanceistypicallycharacterized fortheonesignalthatissupposedtobepresent,weneedamethodtodetermine whathappenswhensmall,contaminating,signalsaccompanythatdesiredsignal.ThisisconsideredinChapter8.Themostimportantnonlinearityinmany applicationsisthatassociatedwiththemixer’sLO;soweemphasizethesystem effectsofcontaminantsontheLO.
Lastly,inChapter9,wewillstudyphasenoise:whereitcomesfrom,howit passesthroughasystem,andwhatareitsimportanteffectsintheRFsystem.
1.3APPENDIXES Materialthatisnotessentialtotheflowofthemaintext,butthatisnevertheless important,hasbeenorganizedin17appendixes.Thesearedesignatedbyletters, andanattempthasbeenmadetochoosealetterthatcouldbeassociatedwith thecontent(e.g.,Gforgain,Mformatrix)asanaidtorecallingthelocation ofthematerial.Someappendixesaretutorial,providingareferenceforthose whoareunfamiliarwithcertainbackgroundmaterial,orwhomayneedtheir memoryrefreshed,withoutholdingupotherreaders.Someappendixesexpand uponthematerialinthechapters,sometimesprovidingmoredetailedexplanations orbackup.Othersextendthematerial.
1.4SPREADSHEETS ThespreadsheetswerecreatedinMicrosoft Excelandcanbedownloadedas MicrosoftExcel97/98workbookfiles(seepagexix).Thismakesthemavailable forthereaders’ownuseandalsopresentsanopportunityforbetterunderstanding. Onecanstudytheequationsbeingusedandviewthecharts,whichappearin blackandwhiteinthetext,incoloronthecomputerscreen.Onecanalso makeuseofExcel’sTracePrecedentsfeature(see,e.g.,Fig.3.5)toillustratethe compositionofvariousequations.
1.5TESTANDSIMULATION Ultimately,weknowhowasystemperformsbyobservingitinoperation.We couldalsoobservetheresultsofanaccuratesimulation,thatbeingonethat
producesthesameresultsasthesystem.Undersomeconditions,itmaybeeasier, quicker,ormoreeconomicaltosimulateasystemthantobuildandtestit.Even thoughtheproofofthesimulationmodelisitscorrespondencetothesystem,it canbevaluableasaninitialestimateofthesystemtobeimprovedastestdata becomesavailable.Onceconfidenceisestablished,theremaybeadvantagesin usingthemodeltoestimatesystemperformanceundervariousconditionsorto predicttheeffectofmodifications.Butmodelingandsimulatingisbasicallythe sameasbuildingandtesting.Theyarethemeansbywhichsystemperformance isverified.Firsttheremustbeasystemand,beforethat,asystemdesign.
Intheearlystagesofsystemdesignweuseageneralknowledgeoftheperformanceavailablefromvarioussystem components.Asthedesignprogresses, wegetmorespecificandbegintousethecharacteristicsofparticularrealizations ofthecomponentblocks.Wemayinitiallyhavetoestimatecertainperformance characteristics,possiblybasedonanunderstandingoftheoreticalortypicalconnectionsbetweencertainspecifications.Asthedesignprogresseswewillwant assuranceofimportantparametervalues,andwemightultimatelytestanumber ofcomponentsofagiventypetoascertaintherepeatabilityofcharacteristics. Finallywewillspecifytheperformance requiredfromthesystem’scomponent blockstoensurethesystemmeetsitsperformancerequirements.
Basedoninformationconcerningthe likelihoodofdeviationsfromdesired performanceprovidedbyoursystemdesignanalysis,wemaybeledtoaccept asmallbutnonzeroprobabilityofperformanceoutsideofthedesiredbounds. Oncethesystemhasbeenbuiltandtested,itmaybepossibletouseanaccurate simulationtoshowthattheresultsachieved,evenwithexpectedcomponent variations,arebetterthantheworstcaseimpliedbythecombinationofthe individualblockspecifications.Tobaseexpectedperformanceonsimulatedor measuredresults,ratherthanonfunctionalblockspecifications,however,requires thatwehavecontinuingcontroloverthe constructiondetailsofthecomponents ofvariouscopiesofthesystem,ratherthanmerelyensuringthattheblocks meettheirspecifications.Forexample, aparticularamplifierdesignmayproduce astablephaseshiftthathasafortuitouseffectonsystemperformance,butwe wouldhavetocontrolchangesinitsdesignandinthatofinteractingcomponents.
Anotherimportantaspectoftestisgeneralexperimentation,notconfinedtoa particulardesign,forthepurposeofverifyingthedegreeofapplicabilityoftheory tovariouspracticalcomponents.Examplesofreportsgivingsuchsupporting experimentaldatacanbeseeninEgan(2000),relativetothetheoryinChapter8, andinHenderson(1993a),relativetoChapter7.Wecanhopethatthese,andthe other,chapterswillsuggestopportunitiesforadditionalworthwhilepapers.
1.6PRACTICALSKEPTICISM Thereisatendencyforengineeringstudentstoassumethatanythingwrittenin abookisaccurate.Thiscomesnaturallyfromourstrugglejusttoapproachthe knowledgeoftheauthorswhosebookswestudy(andtobeabletoshowthison
REFERENCES 5 exams).Withenoughexperienceinusingpublishedinformation,however,we arelikelytodevelopsomeskepticism,especiallyifweshouldspendmanyhours pursuingadevelopmentbasedonanerroneousparametervalueorperhapson aconceptthatappliesalmostuniversally—butnotinourcase.Evenreviewed journals,whichwemightexpecttobemostnearlyfreeoferrors,andclassic workscontainsourcesofsuchproblems.Butthetechnicalliteraturealsocontains valuable,evenessential,information; soahealthyskepticismisonethatleads ustoconsultitfreelyandextensivelybuttocontinuallycheckwhatwelearn. Wecheckforaccuracyinourreferencesources,foraccuracyinouruseofthe information,andtoensurethatittrulyappliestoourdevelopment.Wecheckby consideringhowconceptscorrelatewith eachother(e.g.,doesthismakesensein termsofwhatIalreadyknow),byverifyingagreementbetweenanswersobtained bydifferentmethods,andbytestingasweproceedinourdevelopments.The greaterthecostoffailure,themoreimportantisverification.Unexpectedresults canbeopportunitiestoincreaseourknowledge,butwedonotwanttopaytoo highapricefortheeducationalexperience.
1.7REFERENCES Referencesareincludedforseveralreasons:torecognizethesources,tooffer alternatepresentationsofthematerial, ortoprovidesourcesforassociatedtopics thatarebeyondthescopeofthiswork.Theauthor–datestyleofreferencingis usedthroughoutthebook.Fromthese,onecaneasilyfindthecompletereference descriptionsintheReferencesattheendofthetext.Somenotesareplacedat theendofthechapterinwhichtheyarereferenced.
Practical RF System Design. William F. Egan
Copyright 2003 John Wiley & Sons, Inc.
ISBN: 0-471-20023-9
CHAPTER2 GAIN Inthischapter,wedeterminetheeffectof impedancemismatches(reflections)on systemgain.Forasimplecascadeoflinearmodules(Fig.2.1),wecouldwrite theoveralltransferfunctionorratioas
and u isvoltageorcurrentorpower.Thegainis |g |,whichisthesameas g if u ispower.Thiswouldrequirethatwemeasurethevaluesof u inthecascade. Ifwemeasuretheminsomeotherenvironment,wecouldgetdifferentgains becauseofdifferingimpedancesattheinterfaces.However,itmaybedifficult tomeasure u inthecascade,andagainthatmustbemeasuredinthefinal cascadehaslimitedvalueinpredictingorspecifyingperformance.Forexample, avariationofabout ±1dBinoverallgaincanoccurforeachinterfacewhere thestanding-waveratios(SWRs)are2andachangeashighas2.5dBcanoccur whentheyare3.(SeeAppendixF.1foradiscussionofdecibels(dB).)
Hereweconsiderhowtheexpectedgainofacascadeoflinearmodulescan bedetermined,aswellasvariationsinitsgain,basedonmeasuredorspecified parametersoftheindividualmodules.Throughoutthisbook,gainsandother parametersaresogenerallyfunctions offrequencythatthefunctionalityisnot shownexplicitly.Equationswhosefrequencydependenceisnotindicatedwill applyatanygivenfrequency.
Webeginwithadescription,formodulesandtheircascades,thatapplies withoutlimitationsbutwhichrequiresdetailedknowledgeofimpedancesand
Fig.2.1 Transferfunctionsinasimplecascade.
whichcanbecomplicatedtouse.Thenwewilldiscoverawaytosimplifythe descriptionoftheoverallcascadebytakingintoaccountspecialcharacteristics ofsomeofitsparts.Thiswillleadustoastandardcascade,composedofunilateralmodulesseparatedbyinterconnects(e.g.,cables)thathavewell-controlled impedances.Theunilateralmodules,usually active,havenegligiblereversetransmission.Thepassivecablesarewellmatchedatthestandardimpedance(e.g., 50 )ofthecascadeinterfaces;thesearetheimpedancesusedincharacterizing themodules.
Itiscommontospecifythedesiredperformanceofeachmoduleplusallowed variationsfromthatideal.Thedesiredperformanceincludesagainandstandard interfaceimpedances.Theallowedvariationsaregivenbyagaintoleranceand therequireddegreeofinputandoutputimpedancematches,expressedasmaximumSWRsor,equivalently,returnlossesorreflectioncoefficientmagnitudes (seeAppendixF.2).Thesearetheparametersrequiredfordeterminationofthe performanceofthestandardcascade.Wewillalsofindwaystofitbilateral modulesintothisscheme.
Wewillalsoconsiderthecasewherethemodulesarespecifiedinterms oftheirperformancewithvariousnonstandardinterfaceimpedances(e.g., 2000 – j 500 ),andwewilldiscoverhowtocharacterizecascadesofthese modules.Forcaseswhereitmaybedesirabletoincludethesenonstandard cascadesaspartsofastandardcascade,wewilldeterminehowtodescribe theminthoseterms.
Finally,wewillstudytheuseofsensitivitiesinanalyzingcascadeperformance. ManyvarietiesofpowergainsaredescribedinAppendixG.Ifallinterfaces wereatstandardimpedancelevels(e.g.,50 everywhere),thesegainswould allbethesame,buttheusuallyunintendedmismatchesleadtodifferingvalues forgain,dependingonthedefinitionsemployed.
2.1SIMPLECASES Insomecasesthesecomplexitiesareunimportant.Forexample,whereoperational amplifiers(opamps)areusedatlower frequencies,measurementsofvoltages atinterfacescanbepracticalandtheirlowoutputimpedancesandhighinput impedancesallowperformanceinthe voltage-amplifiercascade toduplicatewhat wasmeasuredduringtest.However,thisluxuryisrareatradiofrequencies.
Modules Cascade
Inothercases,complexitiesmaybeignoredinanefforttogetananswer withminimumeffortorwiththeavailableinformation.Thatanswermaybe adequateforthetaskathand;atleastitisbetterthannoestimate.Commonly, wesimplyassumethatgainswillbethesameaswhenamoduleorinterconnect wastestedinastandard-impedanceenvironment.Wetrytomakethissoby keepinginputandoutputimpedancesclosetothatstandardimpedancewhen designingorselectingmodules.
Whilethissimplifiedapproachcanbeuseful,wewillconsiderherehowto makeuseofadditionalinformationabout modulestogetabetterestimateof cascadeperformance,onethatincludestherangeofgainvaluestobeexpected.
2.2GENERALCASE Tocharacterizethemodulessotheirperformanceinthesystemcanbepredicted, weneedmoreparameters,asetoffour(generallycalledtwo-portparameters;we arecharacterizingourmodulesashavingtwoports,aninputportandanoutput port)foreachmodule(Gonzalez,1984,pp.1–31;Pozar,2001,pp.47–55).We beginbyconsideringtheparametersthatwecanusetodescribethemodules.
2.2.1 S Parameters IndividualRFmodulesareusuallydefinedbytheir S (scattering)parameters (Pozar,2001,pp.50–53;Gonzalez,1984,pp.9–10).Thiscanbedonewiththe helpofthematrix(seeAppendixMforhelpinusingmatrices),
Thesubscriptsinandoutrefertowavespropagating1 intoandoutofthemodule ateitherport(1or2).Theothersubscriptsonthevectorcomponentsindicate theinputport1oroutputport2,whereasthesubscriptoneachmatrixelement isitsrowandcolumn,respectively.Subscript1onthematrixindicatesmodule 1.Weusethesameindexforthemoduleandforitsinputport(port1here). Wecanalsowritethesubscriptsintermsofthesystemwith i or o ,referringto wavestravelingtowardtheinputortoward theoutputofthesystem,respectively. RefertoFig.2.2.Withthisnotation,Eq.(2.3)becomes
Moregenerally,forthe j thmodule,
in,3 = vo,3
out,3 = vi,3
in,4 = vi,4
Bynormalmatrixmultiplicationthen,
Thisisaconvenientformformeasurements.Itrelatessignalscoming“out”of themodule,ateitherport,tothosegoing“in”ateitherport.Wecancontrolthe inputs,ensuringthatthereisonlyonebyterminatingtheporttowhichwedo notapplyasignal,andmeasuringthetworesultingoutputs,oneateachport (Fig.2.3).Thesegiveustwoofthefourparametersandasecondmeasurement, withinputtotheotherport,givestheothertwo. Calibrated generator
Fig.2.3 Measurementsetup.
Thus,formodule1,withport2terminated(vin,2 ≡ vi 2 = 0),wemeasurethe reflectedsignalatport1togivethereflectioncoefficientforthatport,
andthetransmissioncoefficientfromport1toport2,
Thenweturnthemodulearoundandinputtoport2whileterminatingport 1,givingthereversetransmissioncoefficientandport2reflectioncoefficient, respectively:
(Weareusingbothsubscriptformshereasanaidinunderstandingtheirequivalency.)Ineachcasethe S parametersubscriptsrepresenttheportsofeffectand cause,respectively, Seffectcause ,where“effect”istheportwhere“out”occursand “cause”istheportwhere“in”occurs.
2.2.2NormalizedWaves Wehavecalled vx (i.e., vo , vi , vout ,or vin )a“wave,”butthesymbolimplies avoltage.Itiscustomarytousenormalizedvoltageswith S parameters,and theusualwaytonormalizethemisbydivisionoftheroot-mean-square(rms) voltageby √R0 ,where R0 istherealpartofthecharacteristicimpedance Z0 of thetransmissionlineinwhichthewavesreside.Wewillassumethat Z0 isreal.2 AnRFvoltagecorrespondingto vx canberepresentedby
Thiscanbeabbreviated
where
Sometimesaphasorisemployedwhosemagnitudeistheeffective(rms)value (Hewlett-Packard,1996;Yola,1961;Kurokawa,1965):
Ournormalizedvoltage, vx =˜vx / R0 ,(2.16)
usesthisform,whichhastheadvantagethattheavailablepowerinthetraveling wavecanbeexpressedsimplyas px =|vx |2 .(2.17)
Traditionally,thesymbol an isusedfor vin,n and bn isusedfor vout,n . If,ontheotherhand,thephasoremployedinEq.(2.16)is ˆ vx ratherthan vx (Pozar,1990,p.229,1998,p.204),thepowerwillbe |vx |2 /2.Inmostcasesthe moduleparametersareratiosoftwowavesatthesameimpedance;soitmakes nodifferencewhethertheyareratiosof vx orof ˆ vx orof vx
2.2.3 T Parameters Unfortunately,wecannotuse S matricesconvenientlyfordeterminingoverall responsebecausewecannotmultiplythemtogethertoproduceanythinguseful. Werequireamatrixequationforoveralltransferfunctionoftheform
Herethevector Vj ,representingamoduleinput,hasthesameidentifyingnumber (subscript)asthematrix Mj ,representingthemodule.Notethatweareoperating onoutputstogiveinputs.Thisisniceinthatthematricesarethenwritteninthe sameorderinwhichthemodulesaretraditionallyarrayedinadrawing(leftto rightfrominputtooutput,asinFig.2.1).Thereisalsoanevenbetterreason. Thevectoronwhichthematrixoperates(multiplies)mustcontaintheinformation neededtoproducetheresultingproduct. Unilateralmodulesthathavelittleorno reversetransmissiondonotprovidesignificantinformationabouttheoutputto theinput;thusamathematicalrepresentationinwhichthematrixoperatedonthat inputwouldnotworkwell.Ontheotherhand,allmodulesofinterestproduce outputsthatarefunctionsoftheirinputs;sothereissufficientinformationinthe vectorrepresentingtheoutputtoformtheinput.3
Equation(2.18)implies
and
inorderthat
andsoon.Allthisimpliesthat V2 representsthestatebetweenmodules1and2 sowedefinethevector
where j representstheportand o and i indicatethevoltagewavemovingright towardthesystemoutputorlefttowarditsinput,respectively.Thusthematrix connectingsuchvectorshastheform(DechampsandDyson,1986;Gonzalez, 1984,pp.11–12)
Asbefore,themoduleanditsinputhavethesamesubscript.Inmanycasesit willbemoreconvenienttomovethesubscriptfromthevectorormatrixtoits individualelements,addingthe portnumberasthelastsubscript:
Eachvector,inthisrepresentation,describestwowavesthatoccuratasingle pointinthesystemwhereas,forthe S parameters,thevectorelementsrepresented wavesfromdifferentports.4 However, S -parametermeasurementsaresimpler than T -parametermeasurements.Considerthat T121 istheratiobetweenawave enteringthemoduleatport1, vo1 ,andoneenteringitatport2, vi 2 ,while thewaveleavingitatport2, vo2 ,issettozero.Tomeasurethisdirectly,we wouldrequiretwophase-coherentgenerators,onedrivingeachport,thatwould beadjustedsotheoutputsduetoeachatport2wouldcancel.
2.2.4RelationshipsBetween S and T Parameters Itissimplertomeasurethe S parametersandobtainthe T parametersfromthem. Forexample, T22 formodule1is
Equation(2.7)indicatesthatthecondition vo2 = 0requires
CombiningthiswithEq.(2.6)weobtain
fromwhichweobtainthe T parameterintermsof S parameters,
Byasimilarprocesswecanobtaintheothervaluesof Tij intermsofthe Sij :
andof Sij intermsof Tij ,
2.2.5Restrictionson T Parameters Wecannowshowmorespecificallywhythe T matrixwasdesignedtogive inputasafunctionofoutput,ratherthantheconverse.Forunilateralgainin theforwarddirection, S12 = 0.Thissimplifies T22 inEq.(2.30).Ontheother hand,unilateralgaininthereversedirection, S21 = 0,causestheelementsin Eq.(2.30)tobecomeinfinite.As S21 approaches0, V2 becomesaweakfunction of V1 ,soalargenumberisrequiredtogive V1 intermsof V2 .Moreover,if forwardtransmissionissmall, vo2 maybecomeastrongerfunctionof vi 2 than of vo1 ,inwhichcase V1 becomesdependentonthedifferencebetweenthetwo componentsof V2 andsubjecttoerrorduetosmallinaccuraciesin M.Asaresult, M shouldnotrepresentaprocesswheretransmissionfrom V1 to V2 ,asdefined byEq.(2.9),issmallorzero.Forthisreason,Eq.(2.19)iswrittenasitis,since transmissiontowardthesystemoutput S21 isapurposeofasystem,andthusis expectedtobeappreciable,whereasreversetransmission S12 isoftenminimized.
2.2.6CascadeResponse Nowwecanobtaintheoverallresponseofaseriesofmodules(acascade)by multiplyingtheirindividual T matrices.Thesequenceinwhichthematricesare arrayedmustbethesameasthesequence,frominputtooutput,oftheelements inthecascadeandtheinterface(standard)impedancesmustbethoseinwhich the S or T parametersweremeasured.Iftheparametersofadjacentmodules aredefinedfordifferentstandardimpedancesatthesameinterface,oneofthem mustberecharacterized.Thiscanbedonebyinsertinga T matrixrepresenting theimpedancetransition,asdescribedinAppendixI.
