Today’sTopic:MoreLumped-ElementCircuitModels

• Recall:–Wediscussedawire(inductor),resistor(seriesL,parallelRC)lasttime

• Plan:roundoutour“library”ofcomponents– Capacitor,inductor– Examinetheimpactofparasiticelementsoncircuitperformance

• Moveontodistributedcircuits

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

– Easy:justlikeEE20242:V=I*R,V=jwL*I,I=jwC*V– Nothingnew

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

that,butwouldlikeittobesimple– We’llworkupmodelsforcomponents

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

– Fix:modelasinductance.Empiricalformula(veryhandy…)L(µH)=(0.002!)ln(4h/d)

– !=length(incm),d=diameter,h=heightabovegroundplane

ShortWireL(µH)=(0.002!)ln(4h/d)

– !=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,sonotsosensitive)Ø L=7.4nH/cmto12nH/cm

• Doesthismatter?nH seemssmall…• Putthisinacircuitcontext.Assumeh/d=100(12nH/cm)

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

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

OtherComponents:R,L,C• Butfirstsomevocabulary:

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

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

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

• Resistor:lumpedelementcircuitmodel

LumpedElementRmodel• Thismodelisprettygeneral,for!<l/10,butissurprisingly

complexinresponse

LumpedElementRExamples• Small-ish resistor:

– 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,imaginaryparthasbignegativepeakatverylowfrequencies;bigresistorsdon’tworkwellatRF…

LargeResistor– anotherlook• Mag/angleviewoften

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

– Conclusion:bigresistorsdon’tworkwellatRF…

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

• RealpartfallsmoredramaticallythansmallR,lesssothanlarge

• Imaginarypartcomparabletorealpartathighfrequencies

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

• Note:|Z|canbelargerorsmallerthanDCresistance• Notcapturedbyeitherapproximation—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/(wC)– straight-linepartbelow~4MHzorso• Thisbehaviorcanbeaproblemorahelp—butyouhaveto

knowitisthere!

Inductors• Inpractice,inductorsareoftentheleastidealofcommon

passives.

• Performance:L=10µH,C=0.5pF,R=5Ω

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

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

range• PlotXforidealL

(X=wL,L=10µH)andfullmodeltogether

• Matchesonlyatverylowfrequencies

• Bigpeak(inrealandimaginarypart;alsoin|Z|

• Sameformulaasfs forcapacitor,butverydifferentbehavior• BehaviorislousyifyouwantedX=wL• GreatifyouwantaDCshortandRF“open”—calleda“choke”;

probablyactuallymoreuseful…

ImpactonCircuits?• So—doesanyofthismattermuch?Afterall,whatwereally

careaboutiswhetherthecircuitdoeswhatwewantornot• Example:RFlow-passfilter

• Simplepi-networkfilter,easilydesignedusingstandardfiltersynthesistoolsinCADpackages(ADS)

• Valuescomputedautomaticallyfromfilterspecifications

FilterPerformance• Frequencyresponse:RFlow-passfilter

• Nicerolloff,flatpassband,what’snottolike?

RealFilterPerformance• Includethefullmodel

foreachcomponent• Parasitics takenfrom

typicalsurface-mountvalues

• Um…thingsarenotsogood

RealFilterPerformance• Comparison:

• Passband isnarrowerthanbefore—ifwewantedsignalsabove1GHztogetthrough,um…

• “Second”passband at5GHzandabove—ifwewantedtoblocksignalsthere,weblewit

• What’swrong?

Recap:LumpedElementModels

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

• Relieson!<l/10,sonotapropertyonlyofthecomponent,butalsoofthesignals

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

• http://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:

• 0.032pFà 1/wC =50Ω at99.47GHz.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…

DistributedCircuitModels• Sofar:discussed“ideal”and“lumpedelement”models

– “ideal”:! <l/100– “lumpedelement”:l/100<! <l/10– lastoneis“distributed”model:!">l/10– Reminder:there’snofixed“frequency”cutoff—itisalwayssizevs.

frequency• “Distributed”:spatially-varying.Sowe’relookingforamodel

thatexplicitlyincludesthegeometry– Sincethecomponentsareappreciableinsizetowavelength,

propagationeffectsnotonlyimportant;mayevendominate– Wantourapproachtobegeneral,butassimpleaspossible:inherent

trade-offbetweencomplexityandaccuracy– Components:dimensions,materials&properties– Interconnects:dimensions,orientation,proximitytootherelements,

boardandmetalproperties,…

DistributedCircuitApproaches• Thetrade-offbetweenaccuracy&complexityleadsto

multipleapproaches– Fullfieldtheory– Transmissionlinetheory

• Fullfieldtheory:– Sincetheoriginofthedeviationfrom“regularold”circuitdesignis

finitepropagationofelectromagneticwaves:useMaxwell’sequationstoexplicitlyincludepropagation

– Approach:useMaxwell’sequations,boundaryconditions(geometries),materialproperties

– SolveforE,Hfieldseverywhere(twovectorfields—6complexcomponentsateachposition,frequency)

– UseE,Htofindcurrent,voltagevs.position(reduce6componentsto2complexscalars)

– Difficultbyhand;timeconsuming(bycomputer),requiresrealeffort

DistributedCircuitApproaches• Transmissionlinetheory

– Canbeviewedaseither:• Simplificationoffieldtheory,or• Extensionofcircuittheory

– Approach:useintuitiontoreplace“electricallylarge”elementswith“distributedcircuitelements”withknownelectricalcharacteristics.Typicallyconvert2- or3-Dproblemintointerconnected1-Delements

– Muchsimplertocompute:bybasinganalysisonknownstructures,candirectlyfindV,Ivs.position;noneedtocomputeintermediaryfields

– Canoftenyieldintuitiveinsightintocircuitoperation,sinceeachelementhas(usually)relativelysimplebehavior

– But:notrigorous.Reliesondesignerto:• Picktherightcomponenttosubstitutein• Ifcouplingbetweenelementsisimportant,designermustaddthat(orchooseacomponentthathasitbuiltin)

DistributedCircuitApproaches• Howarethesetwoapproachesrelated?

– Fullfieldtheoryisrigorous,allowsevaluationofnewstructuresthatarenotunderstood

– TransmissionlinetheoryelementsaredevelopedtomimictheE&Mbehavioroftypicalstructuresthathaveprovenuseful

Ø Transmissionlinetheorymuchmoreefficient,butmaymislead

• Inpractice,notaneither/orproposition• Commonapproach:

– Usetransmissionlineapproachestofindbehaviorfor“standard”or“simple”partsofacircuit

– Switchtofull-fieldtheoryfortrickyspotsorareasforwhichtheappropriatemodelisn’tclear

– Oncedesignisfinalized,onelastfull-fieldanalysisofthewholethingtoavoidsurprises.Muchbettertofindoutbeforethepartshavebeenbuilt…

FullFieldTheoryApproach• Maxwell’sequations,plusboundaryconditions• AquickrecapofE&M:

∇×E = − ∂∂tB

∇×H =∂∂tD+ J

∇⋅B = 0∇⋅D = ρ

Maxwell’sequations:

B = µHD = εE

Constituitive relations:

Rememberwhatthetermsallmean?