Today’sTopic:ComponentCircuitModels

•  Recall:–  Keydifferencebetweenmicrowave/RFdesignand“regularold”circuitdesigniswhetherfinitepropagaConvelocityoflightissignificantornot

•  Roughidea:–  IfpropagaCondelayincomponentismuchsmallerthanthesignalperiod,then“regularold”approachprobablyok—instantaneousresponse

–  Ifnot,thenwhat?– Andcanwebemoreprecise?

DelayEffects

•  Acouplecircuitstoconsider:

•  Thebasics:KVL,KCL:–  KCL:currententeringandleavinganodeisequal–  KCL:voltagearoundaloopiszero

•  “regularold”designsuggeststhesetwoarethesame:Vout(t)=Vin(t)*R2/(R1+R2)

•  Isthistruefortheoneontheright?–  Yes?No?Maybe?SomeCmes?!SomeCmes.

DelayEffects

•  What’sthedifference?–  ChangesinVin“takeCme”toshowupatVout–  Takes(atleast) ℓ/csecondsforsignaltogofromAtoB(assumingpropagaConatc—aswe’llsee,propagaConiso^enslowerthanthis)

–  Socurrents,voltagesatAandBnotequal•  Thisisonewaytolookattheoriginofinductancefromnon-magneCccircuits

–  But“regularold”circuitdesigntreatsA&Basasinglenode—uhoh

DelayEffects–InPictures•  LookatinputstepfuncCon:

–  Note:propagaConisn’tone-way;signalreachesend,“reacts”toKCL/KVLatnodeB,andthisnewdisturbancepropagatesbacktosource.CanhavemulCplebounces

–  Delayincurrent—lookslikeinductance

So:LayoutMaders

•  Fromtheseideas,canseethatgeometriclayoutofacircuitcanmader—verydifferentfrom“normal”–  Iffrequencyhighenough,orℓlongenough,signalat“A”verydifferentthanat“B”duetothedelay,andbouncesbackandforthtoreachsteady-state

•  “Timedomainpicture”—focusedondelay.Easytounderstand,butrarelyusedinmicrowavedesign

" Instead,we’llmostlyworkinthefrequencydomain(phasors,complexV,I)

•  Timedelayisembeddedinthephase-frequencyrelaCons

FrequencyDomainEffects•  MostcommunicaConsystemssendinformaConusinga

carriersignalwithmodulaCon–  ToroughapproximaCon,signallookslikeasinewave,withsome

wigglesadded–  Afewpictures:

–  Ifvelocity=v,Cmedelay=ℓ/v;wavelengthλ=v/f

FrequencyDomainEffects

•  SosignalissCllasinewave—justhasdifferentphase(andamplitude)atdifferentposiCons–  Delaywillgetsubsumedintothisdependence

•  Givesrisetosomerulesofthumbaboutcomponentsizesandfrequencies–  Acommonone:ifℓ<λ/10,cansCllused“lumpedelement”circuitdesigntechniques

–  OK—butsomecareisrequiredtounderstandwhatthisreallymeans(andwhatitdoesn’t—misusedalot…)

–  Example:considercomponentoflengthℓ=λ/10.End-to-endphasedifference?•  Δ Phase=2πℓ/λ(rad)or360ℓ/λ(deg)•  36°!Notsosmall,certainlynotnegligible

CircuitElementModels•  Soevenifℓ<λ/10,clearly“regularold”circuitdesignisnotok;

36°phasechangesareimportant•  Ifℓ>λ/10,clearlythingsareevenfurtheroff•  Whatcanwedo?Switchdevicemodels.Threelevels:

–  Lowfrequency(ℓ<λ/100):phasedifferencesnegligible:20242OK–  “lumped”circuitmodels(λ/100<ℓ<λ/10):canusemorecomplicated

models,butkeepKVL,KCLaround–  Distributedcircuitmodels(ℓ>λ/10):needsomethingelse—spaCal-

dependentcircuittheory•  Sowhataboutthatλ/10ruleofthumb?

–  Itdoesn’tmeanyoucanignoretheseeffects;meansyoucansimplifyhowyoutreatthem

–  Don’tneedafullelectromagneCcsanalysis,can“patchup”regularcircuitdesignapproaches

–  AlternaCveview:“moresophisCcated”circuitelementmodels(butitissCllreallyjustaband-aid)

CircuitModelsforComponents•  Startwithworkhorsepassives:R,L,C•  Lowfrequencyregime(ℓ<λ/100):

–  Easy:justlikeEE20242:V=I*R,V=jωL*I,I=jωC*V–  Nothingnew

•  “lumpedelement”models(λ/100<ℓ<λ/10)–  Phase/delayisimportant,needtoaugmentourtreatmenttocapture

that,butwouldlikeittobesimple–  We’llworkupmodelsforcomponents

•  Evenawireisn’tsosimple—notanidealshort–  Idealshort:phasedelay=0;wireoflengthλ/10,phasedelay~36°

–  Fix:modelasinductance.Empiricalformula(veryhandy…)

L(µH)=(0.002ℓ)ln(4h/d)–  ℓ=length(incm),d=diameter,h=heightabovegroundplane

ShortWireL(µH)=(0.002ℓ)ln(4h/d)

–  ℓ=length(incm),d=diameter,h=heightabovegroundplane•  Anumericalexample:

–  #22wire(likeforabreadboard):d=25.3mils=0.0643cm(aside:microwavepeopleuse“mils”alot;1mil=0.001”.Yes,inches)–  h/dinrangefrom10/100(insideln,sonotsosensiCve)"  L=7.4nH/cmto12nH/cm

•  Doesthismader?nHseemssmall…•  Putthisinacircuitcontext.Assumeh/d=100(12nH/cm)

–  At10MHz:impedanceofwireisjωL=~j1Ω/cm–  At100MHz:impedanceofwireis~j10Ω/cm–  Dependingonwhattherestofthecircuitlookslike,thiscouldbe

nothing,oritcouldbeabigdeal(isitinserieswith25Ω?Or1000Ω?–  Noteitcanstarttomaderatquitelowfrequencies(below100MHz)

OtherComponents:R,L,C•  Butfirstsomevocabulary:

–  Impedance,admidance,reactance,susceptance—besurewe’reallonthesamepage

•  Z(impedance)=R(resistance)+jX(reactance)•  Y(admidance)=G(conductance)+jB(susceptance)•  Y=1/Z•  Careful:

–  G≠1/R,B≠1/X!–  Probablyobviousifyouthinkitthrough,butsotempCng…

•  Resistor:lumpedelementcircuitmodel

LumpedElementRmodel•  Thismodelispredygeneral,forℓ<λ/10,butissurprisingly

complexinresponse

LumpedElementRExamples•  Small-ishresistor:

–  50Ω–  C=1pF–  L=10nH

~5mmofwireoneachend)

–  Realpartnotchangedmuch,butsignificantimaginarypart

SmallResistor—anotherlook•  Sameresistor,same

data—but|Z|andangle

•  Overallmagnitudestronglyaffected;significantphase

–  Whatyouseedependsonwhatyoulookfor

LargeResistor•  Largeresistor:

–  10kΩ–  1pF–  10nH

–  Realpartfallsoffacliff,imaginaryparthasbignegaCvepeakatverylowfrequencies;bigresistorsdon’tworkwellatRF…

LargeResistor–anotherlook•  Mag/angleviewo^en

easiertointerpret•  |Z|fallingfromshuntC•  Phase–>90°--capacitor

–  Conclusion:bigresistorsdon’tworkwellatRF…

“Intermediate”RExample•  Intermediateresistor:R=100Ω,C=1pF,L=10nH

•  RealpartfallsmoredramaCcallythansmallR,lesssothanlarge

•  Imaginarypartcomparabletorealpartathighfrequencies

“Intermediate”R–anotherlook•  Intermediateresistor:R=100Ω,C=1pF,L=10nH

•  Note:|Z|canbelargerorsmallerthanDCresistance•  NotcapturedbyeitherapproximaCon—needfullmodel•  LifeisnotsosimpleatRF…

Capacitors•  Real-worldcapacitorsaren’tidealeither…

•  Performance:C=0.01µF,L=20nH(1cmofwireateachend)

Capacitors•  Notebigdip(headstozero—hugeholeonlogplot)and

abruptflipinphase

•  Belowfs–reactance<0(likeC);abovefs,reactance>0(L!)•  IdealC:X=-1/(ωC)–straight-linepartbelow~40MHzorso•  Thisbehaviorcanbeaproblemorahelp—butyouhaveto

knowitisthere!

Inductors•  InpracCce,inductorsareo^entheleastidealofcommon

passives.

•  Performance:L=10µH,C=0.5pF,R=5Ω

•  OK…X>0atverylowfrequencies(X=ωL),butnotveryideal

Inductors•  Let’scompare:modelvs.idealL•  Zoominonlow-freq.

range•  PlotXforidealL

(X=ωL,L=10µH)andfullmodeltogether

•  Matchesonlyatverylowfrequencies

•  Bigpeak(inrealandimaginarypart;alsoin|Z|

•  Sameformulaasfsforcapacitor,butverydifferentbehavior•  BehaviorislousyifyouwantedX=ωL•  GreatifyouwantaDCshortandRF“open”—calleda

“choke”;probablyactuallymoreuseful…

ImpactonCircuits?•  So—doesanyofthismadermuch?A^erall,whatwereally

careaboutiswhetherthecircuitdoeswhatwewantornot•  Example:RFlow-passfilter

•  Simplepi-networkfilter,easilydesignedusingstandardfilter

synthesistoolsinCADpackages(ADS)•  ValuescomputedautomaCcallyfromfilterspecificaCons

FilterPerformance•  Frequencyresponse:RFlow-passfilter

•  Nicerolloff,flatpassband,what’snottolike?

RealFilterPerformance•  Includethefullmodel

foreachcomponent•  ParasiCcstakenfrom

typicalsurface-mountvalues

•  Um…thingsarenotsogood

RealFilterPerformance•  Comparison:

•  Passbandisnarrowerthanbefore—ifwewantedsignalsabove1GHztogetthrough,um…

•  “Second”passbandat5GHzandabove—ifwewantedtoblocksignalsthere,weblewit

•  What’swrong?

Recap:LumpedElementModels

•  Havedeveloped“lumpedelement”equivalentcircuitmodelsfortypicalR,L,C,pluswire

•  Reliesonℓ<λ/10,sonotapropertyonlyofthecomponent,butalsoofthesignals

•  Sidenote:beverycauCousofvendorclaims.Theyaren’tlying,butyouneedtounderstandwhattheymean…lookatanexample:

•  hdp://www.usmicrowaves.com/res/ceramic/alumina_ceramic_al2o3_99ghz_thin_film_chip_resistor_re1020t10.shtml

DatasheetDetails•  Here’sthetempingpart:99.47GHz!Thatshouldbegreatformy

mm-wavecircuitat94GHz,right?•  Here’stherealthing:

•  0.032pF!1/ωC=50Ωat99.47GHz.Oh.•  So:at99.47GHz,Z≠50Ω.Z=50Ω||-j50Ω.|Z|=35.4Ω,

ang(Z)=-45°.Ooh.At50GHz?|Z|=44.7Ω,ang(Z)=-27°•  Caveatemptor?Ofcourse…justdotheanalysisfirst,cuta

purchaseordersecond.Theydidn’thideanything…