A Basic Introduction to the gm ID

ABasicIntroductiontotheg/I-BadDesign

mD

Methodology

0.1Abstract

Thisarticleintroducesthereadertotheg/I-bad

mD

designmethodology,whichisawaytohelpCMOS

analogcircuitdesignerslinkphysicaltransistorpa-

rameterstosmallsignalmodels.Itiswrittenatthe

levelofuniversitystudentswhoaretakingafirst

couronanalogintegratedcircuits.Itisalsorel-

evanttoexperiencedengineersinterestedinadesign

flowthatincorporatestechnologydetailsearlyinthe

designcycleandyieldsexcellentagreementbetween

hand-calculationsandcircuitsimulations.

Figure1:Low-levelcircuitimplementationisoften

moredifficultthanhigher-leveldesign.

whytheg/I-badapproachisthebesttoolfor

mD

solvingthisproblem.Next,wewillrehashtheen-

tirediscussionataquantitavelevel.Thiswillentail

areviewoftransistoroperationandachronological

developmentofthetoolswehaveavailable.Finally,

wewillclowithathoroughdesignexample.

0.2Introduction

Followingperhapsalongroadtomaturity,CMOS

hasbecomeanexcellentplatformforanalogcircuit

design.Notonlyisitunrivaledinswitchingand

charge-modeprocessing,butitbenefitsfrompersis-

tentprocessimprovementsfueledbythedigitalcon-

sumermarket.Unfortunately,designersmayfindit

verydifficulttotakeadvantageofthestrengths.A

primaryreasonforthisisthatCMOSbehaviorishard

topredictwithoutusingverycomplexmodels,and

thiscomplexityonlywornswithtechnologyscal-

ing.Designers,incidentallyunderpressuretomeet

deadlines,areforcedtoeitherincorporatecomplex

modelsintotheirhandcalculationsorspiralintoa

Spice-intensivedesignloop.Neitherofthestrate-

giesareaseffectiveorpleasantaswewouldlike.

0.3TheBigPicture

0.3.1AnalogDesignReliesonAb-

straction

Fig.1showsverallevelsofabstractioninwhichwe

canviewananalogdesign.Thankstoabstractions,

Mygoalinthisarticleistointroduceyoutothe

engineersworkingatthehigherlevelscanperform

g/I-baddesignmethodology,whichgreatlyim-

mD

analysisusinglinearSignals-and-Sytemstheory.This

provesthepredictabilityofCMOSsmall-signalbe-

isthedomainoffilters,gainblocks,OpAmpcircuits,

haviorwithoutrequiringcomplexequations.Wewill

etc.Themathematicsthatgovernthisrealmareel-

definetheratiog/Iinmoredetaillater,butfor

mD

egant,oftenwithcenturies-oldroots.Conquently,

now,justthinkofitasadesignvariablethatencap-

wehavegottenverygoodatunderstandinghowto

sulatesthebiasingconditionsofaMOStransistor.

workwiththeblocks.Mostengineeringschools

Or,evenmoreconcily:

ndstudentsthroughanentirebatteryofcours

thatsatisfactorilycoverthisarea.

g/I≈biaspoint󰀙small-signalmodel

mD

Descendingtothelowerlevels,thereisadifferent

story.Whilewefinditstraightforwardtobuilda

Developmentofthemethodologywillinvolveveralgain-of-twostageusinganOpAmp,wefinditvery

steps.WewillstartwithaverybroadoverviewofdifficulttobuildtheOpAmpitlf.Howbigshould

analogcircuitdesigntoewhatproblemitisthateachtransistorbe?Howmuchbiascurrentisneeded?

weareattempingtosolve,andhowithasbeensolvedThelow-leveldecisionscanbeunclear,andthere

inthepast.Iwillthenexplain,ataqualitativelevel,aretwobigreasonswhy.First,transistorbehavior

1

Figure2:Small-signalmodelofatransistor.

isnonlinear,andclassicalSignals-and-Systemsanal-

ysfallapartwhenappliedtononlinearsystems.

Second,technologyadvancementschangetherulesof

thegamefasterthanwecanmakethem.Thereare

nocenturies-oldrootshere!Asaresult,wesimplydo

nothaveanicetoftransistorequationsthatisboth

compactenoughforhandcalculationsandaccurate

enoughtomatchSpicesimulations.

Figure3:Smallsignalmodelsallowustoutran-

sistorsinaSignals-and-Systemscontext.

0.3.3Whyg/IisBetterthanV

0.3.2MakingLow-LevelDesignMan-

mDov

V-baddesign,whichwewillshortlycoverinmore

ov

ageable

detail,longpredatesg/I-baddesign.Aswehave

mD

alreadyhinted,bothVandg/Iarequantities

ovmD

thattellyousomethingaboutthebiaspointofatran-

sistor.So,howaretheapproachesdifferent?

WhenCMOSdesignerschootofollowaV-badAsyouknow,wedothisbyapproximatingeachtran-

ov

designstrategy,theyimplicitelyacceptthevaliditysistorwithafewidealelements,collectivelyreferred

ofthelong-channelmodel.Iamcertainthatyouaretoasasmall-signalmodel.Fig.2showsabasicand

familiarwiththelong-channelmodel(wewillalsofamiliarsmallsignalmodelofaMOSFET.Italso

reviewitinalaterction).Whenwewerefirsthighlightsthetranslationalrolethatg/I(orits

taughthowtoanalyzeaMOSFET,wewereshownpredecessor,V,anotherbiasingvariable)playsin

aderivationofitusingbasiccalculus.Unfortunately,thedesignprocess.Ofcour,thedrawbackofusing

mostoftheassumptionsthatmakethederivationsosmall-signalmodelsisthattheyintroduceerrors,as

cleanareuntruefortoday’ssmallgeometries.Con-allapproximationsmust.Butthatisfaroutweighed

quently,theV-badmethodologynolongeryieldsbythebenefitsofusingSignals-and-Systemstech-

ov

circuitsthatbehaveasintended.Inordertosalvageniques,withoutwhichwewouldnothaveconcepts

themodel,designershavetriedtopatchitwithshort-likegain,bandwidth,frequencyrespon,poles,and

channeleffectsandavarietyofcurve-fittingtermszeros!

thatare(sometimesonlywishfully)badondiffer-

entphysicalarguments.Butintheend,V-bad

ov

designonlygetsharderandlessaccurate.

Ournewstrategy,g/I-baddesign,doesnotrelywhereweareverycomfortable.Atthebottomare

mD

onthevalidityofthelong-channelmodel.Infact,physicaltransistors,which,intheend,mustbehave

itdoesnotrelyonthevalidityofanythingexceptthewaywewantthemto.Sittinginthemiddleofall

simulation.Thismethodoloyislookup-table-bad.thisisg/I,anintermediatebiasingvariablethat

Theunderlyingphilosophyisthattheequationsgov-bridgestheabstract-to-physicalgapverywell.Keep

erningMOSFETsaresocomplexthatwemustgetthispictureinmindaswecontinueourdiscussion.

2

Wecanmakelow-leveldesigneasierifwetransform

transistorsintoSignals-and-Systems-friendlydevices.

mD

ov

Fig.3,then,isagoodillustrationofhowtheg/I-

mD

baddesignmethodologyfitsintothebigpicture.

AtthetopistheabstractSignals-and-Systemsworld,

mD

0.4TheMore-DetailedPicture

NowIwantustostartovertryingtosolvethedesign

problem,butatamorequantitativelevel.Wewill

reachthesameconclusion,ofcour,eventhoughwe

aretakingaverydifferentapproach.

AFirstAttemptatTransistor-LevelDesign

Howmightanintelligent-but-inexperiencedengineer

goaboutdesigningacircuit?Ofcour,Ihavea

preferredmethodtowardswhichIamworking,but

itiscertainlyworthwhiletoeifwecansolvethe

designproblemwithoutknowingtheansweraheadof

time.

Tobegin,letusstepbackandask,whatwillour

finisheddesignlooklike?Or,whatisafinishedde-

sign?Inthecontextofthisarticle,itisanetlist.

Ultimatelywejustwantafilethatcontainsspecifica-

tionsforallthetransistors,resistors,capacitors,etc.,

andexplainshowtheyareallconnectedtogether.Of

cour,intherealworld,circuitsmustbefabricated,

anddesignersmustbewaryofthelimitationsofsimu-

lationitlf,andhowwellitagreeswithactualmea-

suredperformance,butthoconcernsarebeyond

ourscopehere.

Figure4:High-levelcomparisonoftwopopularde-

signmethodologies.

ridoftheminfavorofafewtablesorgraphs.And

becauthegraphsaregeneratedusingdevicesim-

ulationsinSpice,theyaremuchmoreaccuratethan

thelong-channelmodelcouldeverhopetobe.

Ifourendgoalisanetlist,whynotstartwiththe

Itisnostretchtosaythatmostofuscringealittleat

netlistandworkbackwards?Whatkindsofinforma-

thethoughtofusingalookuptable.Withtheadvent

tiondoweneedinorderto”fillintheblanks?”For

ofcheapandpowerfulcomputing,weelectricalen-

reference,hereisalinethatinstantiatesatransistor

gineershavelosttouchwithfiltertables,logtables,

inHspice:

trigonometrictables,andthelike.Butourcurrent

equationalexclusivityhasbeenonlyabrieffadin

M1drngatsrcblknchmodelL=0.18uW=10u

ourindustry.Justasvacuumtubecircuitdesigners

Well,whichblankscanwefillin?Putanotherway,

onceud(andstillu!)tubecurves,soweareredis-

howdowedesignatransistor?Obviously,V,µ,C,

Tox

coveringthevalueoftable-baddesignforsituations

andotherfamiliartransistorquantitiesarenotamong

whereitisthemostefficientmeansof”computation.”

theparameterswegettospecify.Infact,apartfrom

Fig.4comparesV-baddesignandg/I-bad

ovmD

designinaside-by-sidesummary.Inbothcas,we

needphysicalinformationaboutthetechnologytar-

get.Afterall,thecapabilitiesofthetargetwillob-

viouslyaffecttransistorperformancegreatly.Inthe

caofthelong-channelmodel,thetechnologydata

mustbelimitedtoonlythebarestofesntials,such

asµandC,otherwihandcalculationsbecomein-

ox

tractible.Conquently,initialdesignsmayonlyget

withinanorderofmagnitudeuntilthedesignergetsa

”feel”forthatprocess.Meanwhile,theg/I-bad

mD

methodutilizescompleteSpicemodelsfromthetech-Onepossibledesignmethod,then,mightbetojust

nologytargetandyieldsinitialresultsthatonlyre-uWandLdirectlyasdesignvariables.Thispro-

quireminortweaking.cesswouldbesomethinglikethefollowing:

3

theterminalconnections,itlookslikeweonlygetto

chooWandL.

Isthatallthereistoit?Iscircuitdesignjustamatter

ofdecidinghowbigeachtransistoris?Well,yesand

no.Withtheexceptionofsomeadvancedoptions

(suchassourceordrainsharing,ormulti-fingered

gates),WandLreallyaretheonlytransistorcharac-

teristicsthatyougettoexplicitelyspecify.Youhook

themtogether,sizethemcorrectly,andyoualmost

havethewholething.Really!

1.Assumeyouhaveausabletopology.

2.GuessabunchofvaluesforW’sandL’s(and

possiblyR’sandC’s).

3.SimulateinSpice.

4.Seeifthedesignmeetsallthespecifications

5.Ifnot,modifyW’sandL’s(andpossiblyR’sand

C’s)andgobacktoStep3.

Note:Thiskindofiterativeprocessissome-

timescalled”SpiceMonkeying,”anditsu

isstronglydiscouraged.Itisverycommon

fordesignersofallexperiencelevelstolap

intoSpiceMonkeyinginthefaceoflooming

deadlines!

Thisstrategy,howevertempting,doesnotworkwell

inpractice.First,thesheerlengthoftimerequired

forsimulationmakesitimpracticaltoruntoomany

ofthem(andthisistrueevenifyouautomatethe

process).MoreimportantlyaSpice-intensivedesign

methodisablindtrek,completelydevoidofintuition.

Spiceisgoodforanalyzingadesignandmakingfinal

tweaks,butitisnotverygoodathelpingyoudecide

amongtheinfinitudeoftopologies,sizings,andbias

pointsavailableinanopen-endedtask.

Figure5:Wecandomoreeffectiveoptimizationin

thehand-calculationsphathaninthesimulations

phaofthedesignprocess.

ofthemostcommonlyudtodisplaytherelation-

ships.Onethingthatisalittledifferentaboutthis

plot,comparedtoothersyoumayhaveen,isthe

uofVinsteadofV.Viscalledtheoverdrive

ovgsov

Allthisissummarizedinthesimplifieddesignflow

voltage,anditisdefinedasfollows:

showninFig.5.Theinner-mostloop,loolycalled

the”handcalculations”pha,iswherewehavethe

bestopportunitiestomakebig-impactdecisions.If

V=V−V

ovgsT

wehavemeaningfulandaccuratesmall-signalmodels,

wecanmakeinformedandconfidentdecisionsinthis

innerloopwithoutresortingtofrequentsimulation.

Letuseifwecandevelopareliablelinkbetween

Vtellsyouhowinvertedthechannelis,andisa

ov

small-signalmodelsandactualtransistorbehavior.

littleeasiertoworkwiththanV,inpartbecau

gs

ithidesanydependenceonV.Inversion,so-called

T

becauthematerialinthechannel(e.g.p-type),

0.4.1Long-ChannelModelReview

startstobehaveliketheinvertypeofmaterial(e.g.

n-type),canberoughlyinterpretedas”ON-ness.”It

Thebestvehicletocarryusfurtheronourquanti-

isbecauitcontrolsthelevelofinversionthatwe

tativediscussionofMOSFETbehavioristhelong-

canconsiderVtobeabiasingvariable.Sometimes

ov

channelmodel.Ofcour,Ijusttoldyouthatthis

theconditionofhavingaverysmallVisreferredto

ov

modelwasinadequate,but,justtobeclear,Iamnot

asweakinversionwhilealargeVmaycaustrong

ov

advocatingitscompleteabandonment.Thederiva-

inversion.

tionmaybeover-simplified,butitstillusuallygives

therightkindofintuition;andwedonotwanttobeAlsodenotedinFig.6arethethreeoperatingregions:

robbedofthat.Inaddition,itissimplyagoodplacecutoff,linearandsaturation.Wewillquicklygoover

eachone.tostartwhendiscussingtransistormodelling.

Thelong-channelmodelattemptstodescribethere-Note:ThisisanN-Channel-centricreview.Youwill

lationshipsbetweendrain-current,I,andthetermi-havetoapplytheusualflipstogettheP-Channel

D

nalvoltages,VandV.TheplotinFig.6isonerelationships.

dsgs

4

Figure6:Inthesaturationregion,Iisprimarilya

D

functionofV.

gs

CutoffRegion

ConditionV<0,(or,equivalently,V<V)

ovgsT

Thereisnochannelinversion,sonocurrentflows

I=0

D

LinearRegion

ConditionV≥0andV<V

ovdsdsat

Thereischannelinversion,butIisheavilyaffected

D

byV

ds

I=µC2VV−V

DoxovDS

󰀃󰀂

1W

2

DS

2L

Figure7:Asaturation-centricviewoftransistorbi-

asing.

Saturationisthedesiredoperatingregionformostof

thetransistorsinthesignalpath,otherthanswitches.

Infact,fortherestofthisarticle,wewilloper-

atealmostexclusivelyinthesaturationregion.As

longaseachtransistorhasenoughheadroom,mean-

ingthatwemaintainV≥V,thenwecanadopt

dsdsat

asaturation-centricpointofview,whichisshownin

Fig.7.Notethat,asVincreas,notonlydoesI

ovD

increa,butggrowsaswellduetothequadratic

m

equation.Inotherwords,gisafunctionofV.

mov

Keepthispictureinthebackofyourmind.

0.4.2IntroductiontotheV-Bad

ov

DesignMethodology

Thelinearregionisalwaysofconcernwhenthetran-

sistorisbeingudasaswitch.Inthatca,youcan

alsodefineitsON-resistance.

R≈

on

∂V

ds

∂I

D

g,fandMakingSenofTransistor-Level

mT

Design

NowthatwehavebeenintroducedtoV,wecande-

ov

velopitintoadesignvariable.Remember,weeven-

tuallyneedittotieintothesmallsignalmodelshown

inFig.2.Thefirstelementinthemodelwewillwork

onisg,whichisjusttheslopeoftheIvs.V

mDov

curve.

g==µCV

moxov

W∂I

D

∂VL

ov

SaturationRegion

ConditionV≥0andV≥V

ovdsdsat

Ibecomespurelyafunctionofthegatevoltage,V

Dov

(notofV)

ds

I=µCV

Dox

1W

2

ov

2L

5

Withalittlealgebraicmanipulation,wecanderive

aninterestingequation,which,asyoumayrecognize,

containsbothofthebiasingvariablesthatweare

investigating.

2g

m

=

IV

Dov

Ibringthisupnowfortworeasons.First,itgivesyou

anideaoftherelationshipbetweeng/IandV.

mDov

Wewillelaterthatthisequationisinadequate,

becautheactualrelationshipbetweentheterms

ismuchmorecomplex.Butfornow,justkeepin

mindthatthetwoquantitiesaresomewhatsimilar.

Acondreasonforbringingupthisequationisto

definetranconductorefficiency,whichisjustanother

wordforg/I.Thistitleisappropriateifyouun-

mD

derstanditsmeaning.Iliketogiveittheintuitive

unitsofmS/mA(ratherthansimplifyingto1/V)be-

cauitcapturesthespiritoftheterm.Ittellsyou

howmuchg(inmS)yougetwhenyouinvesta

m

givenI(inmA).

D

Goingbacktogitlf,letuslookatsomeexpres-

m

sionswecanderiveforitandeifwecanmakeany

noutofthem.Hereiswhatweknowsofar:

g=µCV=

moxov

W2I

D

LV

ov

Figure8:Examplecircuit.

g=2I/V==6.7

mDov

mA2·1mA

300mVV

mAVv

out

=−6.7·1kΩ=−6.7

vVV

in

Infact,wecanuItomakethegainwhateverwe

D

want.Suppowedoubleit.

g=2I/V==13.4

mDov

2·2mAmA

300mVV

Whatdoesthismean?IsgisproportionaltoI?

mD

Ifyouwantmoretransconductance,doyouhaveto

investmorecurrent?Thatemsreasonable.But

howaboutV?WellnowIknowweareintrouble

ov

becauitisinthenumeratorinonecaandinthe

denominatorintheother!

Inordertomoveforward,letusassumethatwecan

programVtoanyconstantwewant.Furthermore,

ov

wearegoingtothinkofVasaknobthatwecan

ov

utodeterminetransistorbehavior.Wecouldhave

chonadifferentknob,likeIorW/L,butVwill

Dov

turnouttobeabetterknobthanthe.

LetusdoanexampletoshowwhyaconstantVis

ov

anicething.InFig.8,wewilloperateM1withaV

ov

of300mV.Thisisareasonablevalue,anditisalso

anarbitrarychoice.Also,arbitrarily,suppothat

I=1mA.Howcanweanalyzethiscircuit?

D

M1has1mAofdraincurrent,andsuperimpodon

thisissomesignalcurrent,i,whichisafunction

s

ofv.Whatkindoffunction?Weareassuminga

in

linearonebadong.Infact,weexpectsomething

m

likethis:

v=−i·R=−v·g·1kΩ

outsinm

AndbecauweknowVandI,weknowgignoringspeed.Howfastisourtransistor?Canwe

ovDm

6

vmAV

out

=−13.4·1kΩ=−13.4

vVV

in

Whatabreeze!OncewemakeVaconstant,wecan

ov

programthegainbyadjustingI.Analogdesignin

D

CMOSissoeasy.

Well,notquite.Thoughlaterwhenwegetfinished

withg/I-baddesign,youmightactuallyfeelthis

mD

wayalittlebit!Butwearenotthereyet.Thereis

somethingveryobviouslywrongwithwhatwehave

done.Youmayhavecaughtifyouarepayingatten-

tion.IfwecanmakeVwhateverwewant,then

ov

whynotmakeitzero?Isthatnottheidealvalue?I

mean,ifthefollowingrelationshipistrue,

g=

m

2I

d

V

ov

thenzeroVwouldmakethecircuitinfinitelyeffi-

ov

cient!Certainlyyoursuspicionsarearoud,asthey

shouldbeanytimeyoumanagetomakeacircuitin-

finitelyefficient.Wemustbedoingsomethingwrong,

butwhat?

Itisactuallysomethingverybasic.Wehavebeen

Figure10:Vislikeagvs.fknob.

ovmT

Figure9:Vcontrolsthetradeoffbetweentranscon-

ov

ductorefficiencyandf.

T

putspeedintermsofV?Theanswer,ofcour,is

ov

yes,andthederivationispainless.First,wedefine

󰀄󰀅

f=

T

1g

2πC

m

gs

Thisisoftencalledthetransitfrequency,anditisthe

answertothequestion”Howfastisthistransistor?”

Insaturation,C=CWL.Ifweplugthatback

gsox

2

3

intothelong-channelmodel,weget

Figure11:DesignexampleshowinghowtouV.

ov

f==

T

1g12µV

mov

2πC2π3L

gs

2

highVandlivewithlowertransconductorefficiency

ov

Well,guesswhat.Nowweknowwhywecannotmake

(i.e.highpower).

Varbitrarilysmall–becauitlimitsspeed.We

ov

Thisreallyistheheartofthematter.Visu-

ov

havefoundagvs.ftradeoff,whichisillustrated

mT

fulprecilybecauitletsyoumanagethetradeoff

inFig.9.

betweentwoofthethingsyouneedmostinanalog

LetusexamingFig.9morecloly.Earlier,wede-

cidedthatVwasaknobthatwecouldutoadjust

ov

transistors.Atthetime,Itoldyouitwasanarbi-

trarydecision,butnowperhapsitisstartingtolook

likeitwasalsoagoodone.Twothingsthatwecare

about,gandf(whichcouldroughlytranslateas

mT

gainandbandwidth),arebothdependentonVin

ov

verysimpleways.Evenbetter,thetwothingshave

conflictinginterestsregardingV,whichmeanswe

ov

canfindanoptimumandmakedesignchoices.For

example,ifwecanlivewithaslowdesign,thenwe

canualowV,whichwillyieldahightranscon-UsingthecircuitinFig.8,suppowewant500MHz

ov

ductorefficiency(i.e.lowpower).Ontheotherhand,bandwidthandwewantagainof10.Onepossiblede-

ifthecircuitneedstobefast,wemustoperatewithasignflow,listedhere,canbefollowedalonginFig.11.

7

design.ThinkofitlikethescrewdriverinFig.10.For

afixedcurrent,I,wecanuVtodecidewhether

Dov

wewanttospendthatcurrentinvestmentong(to

m

getmoregain)oronf(togetmorebandwidth).

T

Nowwewanttoplugallthisbackintothedesign

flow.RememberFig.2.Wewanttobeableto

approximateatransistor,whichisnonlinear,witha

smallsignalmodel,whichwecanuinSignals-and-

Systemsanalysis.Wecanillustratehowtodothis

withanexample.

1.Toachievetherequiredgain,wemusthavereliable-evenifwepatchthemodelwithextraGreek

g=10/1k=10mA/V

m

V

V

letters.Nowitistimetoprentsomehardevidence.

Thefirstbitofevidenceissimplecommonn:

WhydoyouthinkSpiceusdozensoftransistor

parametersduringsimulation?Doyouthinkthat

itignoresallbut2or3ofthem?Ofcournot!

Conquently,weshouldbeshocked(andmodelling

engineersshouldbeembarrasd)ifµandCproved

ox

tobeascompleteastheentireSpicemodel.

Figs.12and13illustratethisgraphically.InFig.12,

wecompareactualtransconductorefficiencysimu-

latedinSpicetothelong-channelprediction.For

largevaluesofV,thelong-channelmodelisonlyoff

ov

by25%-nottoobad.ButforsmallV,thevalues

ov

arenonn-mostnotablyatV=0,whereitstill

ov

insistsoninfinitetransconductorefficiency.Fig.13

showsthatthelong-channelmodeldoesnobetter

inpredictingf.Andtheinterctionwhereboth

T

graphsareontargetishardlyexistentatall.

2.Andforaninputpoleat500MHz,weneed

C==1.1pF

gs

1

2π×300Ω×500MHz

3.Wecantheneasilycalculatetherequiredtransit

frequency

f===9.4GHz

T

g10mS

m

C1.1pF

gs

4.Whichmeans(eFig.9,Note4)

V≥75mV

ov

5.Whichmeans(eFig.9,Note5)

g/I≤26mS/mA

mD

YoumayalsohavenoticedthatFigs.12and13con-

tainaregionofnegativevaluesforV,whichisalso

ov

6.Andfinally

calledthesubthresholdregion.Thelongchannel

g10mS

m

modelpredictsthatforV≤0,I=0,implying

ovD

I===385µA

D

g/I26mS/mA

mD

weshouldneverbiasaMOSFETnearVbecauit

T

mightturnoff.ButthesimulationdatainFig.12

Prettyeasy,really.Notonlythat,butweknowthat

showsthatg/Iactuallycontinuestoclimbaswe

mD

ourdesignisefficient.AlargerVwoulddelivera

ov

headintothesubthresholdregion,whichmeansthat

faster-than-necessarytransistor,andwewouldwaste

thismayreallybeaufulbiasingpoint.Isit?

power.AsmallerVwoulddeliveraslower-than-

ov

Infact,thesubthresholdandweakinversionregions

requiredtransistor,andwewouldnotachievethede-

areveryimportantinlow-powerdesigns.Fig.13ex-

signgoals.

plainsthetradeoff,whichisthatsubthresholdtran-

Notethatrhasnotbeendetermined,sotechnically

o

sistorsareslow.Butkeepinmindthattheyarenot

thesmallsignalmodelfromFig.2isalittlelacking.

thatslowinnewertechnologies.Infacttheycanoften

Inthisarticle,willnotdealwithrinaVcontext.

oov

bemuchfasterthanweneedthemtobe,whichallows

Later,oncewehaveintroducedg/I-baddesign,

mD

designerstotradesomeofthatspeedforlowpower.

Formanyoftoday’spower-constraineddesigns,sub-

wewillinvestigatethelimitationsduetor.

o

thresholdoperationisimperative,andamodelthat

Insummary,findinganoptimumoperatingpointus-

doesnotaccomodatethisregionisuless.

ingtheV-baddesignmethodologyismuchmore

ov

straightforwardthanworkingbruteforcewithWor

Whatdoesallthismean?Itmeansthat,althoughV

ov

L.AndthinkingintermsofVmakesiteasyto

ov

isaverygooddesignvariableintheory,itdoesnot

dealwiththeinterdependenciesofg,Candbi-

mgs

workinpractice.Ifthelongchannelmodelwereac-

asingthanbytryingtomanagetheindependently.

curate,thenV-baddesignwouldworkbrilliantly

ov

Theonlyrealproblem,aswepointedoutearlier,is

andtherewouldbenoreasontolookelwhere.But

thelong-channelmodeldoesnotworkandsoweneed

thatthelong-channelmodelisnotaccurate.

anewdesignvarible:somethinginthespiritofV,

ov

butthatyieldsbetteragreementbetweenhandcal-

0.4.3TheLimitationsofV-Bad

ov

culationssimulation.

Design

Wehavebeenhintingallalongthatthelong-channel

modelisjustwaytoosimpletomakeV-baddesign

ov

8

Figure13:Vdoesnotpredictfverywell.

ovT

Figure12:Vdoesnotpredicttransconductoreffi-

ov

ciencyverywell.

0.4.4Introductiontotheg/I-Bad

mD

DesignMethodology

g/IasaDesignVariable

mD

Forstarters,suppowetriedtokeepVasadesign

ov

variable.WecouldusimulationstoehowV

ov

affectsbothtransconductorefficiencyandf,similar

T

towhatwasdoneinFigs.12and13.Usingthotwo

charts,wecancertainlyfindthebestvalueforVto

ov

suitourneeds.

Fortunately,someonehasalreadycomeupwitha

muchbetterideathatencapsulatesallthisinforma-

tioncompactly.RecallthatVandg/Iaresimi-

ovmD

lar,biasing-relatedquantities.Iftheyaresosimilar,

maybewecangetridofoneofthem.Theclever

trickistoplotfvs.g/Idirectly,asshownin

TmD

Fig.14.Thisplotletsyoueexactlyhowincreas-

ingtransconductorefficiencycomesatthecostoff.

T

WearecuttingouttheVmiddle-man,sotospeak.

ov

Fromhereon,g/Iistheonlybiasingvariablewe

mD

willneed.

Bytheway,whereisthesubthresholdregionin

Fig.14?YoucangetahintbylookingatFig.12.

Thesubthresholdregioniswhereweexpecttofind

thehighestvaluesfortransconductorefficiency.In

otherwords,itliessomewhereonthefarrightsideof

Fig.14.Doesitmatterexactlywhereitbegins?Of

cournot!Weknowg/I.Weknowf.There-

mDT

gionweareinisirrelevant.Infact,withtheexception

ofthelinearregion,ofwhichwemustalwaysbecare-

9

Figure14:Amoredirectdepictionoftheg/Ivs.

mD

ftradeoff.

T

ful,alltheregionsofoperationbecometransparent

whenweutheg/I-baddesignmethodology.

mD

Again,letusdoanexamplebadonthecircuitin

Fig.8.And,again,assumethatwewantasignal

bandwidthof600MHz,againof10,andthatwe

wanttouaslittlepoweraspossible.Wecanfollow

alonginFig.15todothedesign.

1.Toachievetherequiredgain,weagainmusthave

g=10/1k=10mA/V

m

V

V

2.Andforaninputpoleat500MHz,westillneed

C==1.1pF

gs

1

2π×300Ω×500MHz

Figure16:Anfdesignchartfora0.18µmprocess.

T

Figure15:Usingg/Iinasimpledesign.

mD

3.Sothetransitfrequencymustbe

f===9.4GHz

T

g10mS

m

C1.1pF

gs

4.Whichmeans(eFig.15,Note4)

g/I≤17.5mS/mA

mD

5.Andfinally

I===570µA

D

g10mS

m

g/I17.5mS/mA

mD

Whataboutr?

o

Easy.Ofcour,sowastheV-badmethodwe

ov

lookedatearlier.Theimportantdifferenceisthat

Fig.17explicitelyhighlightsr’splaceinatransistor

o

circuit.Asyouknow,rissimplyanotherloadin

o

thenewrelationshipsareaccurate.

parallelwithR.Inpractice,weusuallyeitherwant

L

Fig.16isamorecomplete(andactuallyourfinal)

r󰀆Rsothatrcanbeignored,orwewillmake

oLo

versionofthefvs.g/Idesignchart.Itincor-

TmD

Rverylarge(perhapsusingacurrentsourceinstead

L

poratestheeffectofchannellength,L,ontransistor

ofanactualresistor)inordertogetasmuchgainas

speed.Severallengthsareincluded,rangingfromthe

wecan.Iwanttoexaminethecondca.Letus

processminimumuptoareasonablylargevalue.We

ewhathappensasR→∞.

L

willexplainhowtocreatethischartinalaterction.

Fig.18illustratesthiscaeffectively.Whenrbe-

o

LetuslookclolyatFig.16.Foranygivenvalueof

comesthedominantresistiveload,theoverallgain

g/I,alargerLalwaysmeansaslowertransistor.

mD

islimitedbywhatwerefertoastheintrinsicgain

Thismeansthat,ifwehavenootherconstraints,we

ofthetransistor.Intrinsicgainistheproductofg

m

shouldalwayschootheshortest-lengthtransistor

andrandtellsusthehighest(Voltage)gainwecan

o

available(i.e.theprocessminimum).Italsomeans

possiblyget.

thattheremustbesomeotherconstraintwemight

needtoconsider.Thatconstraintisr,andwewill

o

󰀆󰀁

IntrinsicGain=grcoveritnow.

mo

V

V

10

Figure17:rofacommon-sourceamplifierappears

o

asanadditionalload.

Figure18:Whatisthemaximumgainwecanget

fromatransistor?

Figure20:Abiasingchartfora0.18µmprocess.

Asanexample,suppoyouwouldlikeacircuitwith

againofatleast50.Aquickglanceattheintrin-

sicgaincharttellsyouthatthisiscertainlyfeasible.

Choosingtheminimumchannellength,L=0.18µm,

isprobablytooriskysinceitleavesnomargin,but

perhapsL=0.28µmwouldbeconrvativeenough.

Now,whatifwewantagainof100?Thistimea

quickglanceatthechartsaysthatwecannotgetthis

Figure19:Anintrinsicgainchartfora0.18µmpro-

fromasingletransistor.Wemightneedtoeitheru

cess.

multiplestages,orperhapstryacascodecircuitin

ordertoincreathegain.Thoareproblemswe

cansolve.Theimportantthingisthatweknowthis

Inpractice,intrinsicgainisamoreconvenientnum-stuffnow.Wedonothavetowastetimetryingto

bertoknowthanritlf(wewillewhyshortly).makethedesignworkwithasingletransistor,only

o

Mathematicallytheyarequasi-equivalent.Thatis,tofindoutthroughveralSpiceiterationsthatit

youcanputgains”inparallel”justlikeyoucanputcannotbedone.

R’sinparallel.Forexample,ifacircuithasanideal

gainof10,andyouuatransistorwithanintrinsic

Insummary,thetwochartsgiveafairlycomplete

gainof100,thenthenetgaindegradesbyabout10%.

pictureoftransistorbehaviorbadong/Iasa

mD

designvariable.Weknowhowbothfandgr

Tmo

Wecanutheconceptofintrinsicgaintocreateareaffectedbybothg/IandL.Becauofthis

mD

anotherdesignchart.Fig.19issimilartoFig.16,wecanchoothebestLandg/Iforthejob.

mD

exceptthatintrinsicgainisthedependentvariable.Andbecauthechartsaresimulation-bad,weare

LikeFig.16,itisplottedvs.g/I,andthatisdoneconfidentthattheyareaccurate.

mD

forveraldifferentchannellengths.

Asmallwarningisinorderbeforewemoveon.Be

Figs.16and19,takentogether,compriaverypow-awarethatrisverydependentonV.Thevalues

ods

erfuldesigntool.NotonlyaretheyextremelyhelpfulprentedinthechartsareforV=V/2.IfV