Design Methodology for MFB Filters in ADC - Texas Instruments

32 Pages·2006·1.45 MB·English

Application Report
SBOA114–February2006
Design Methodology for MFB Filters in ADC Interface
Applications
MichaelSteffes.................................................................................................... High-SpeedProducts
ABSTRACT
While the Multiple FeedBack (MFB) filter topology is well-known, its application to very
high dynamic range analog-to-digital converter (ADC) interfaces requires a careful
consideration of component value selection. This application report develops the ideal
transferfunction,thenintroducesacomponentselectionmethodology and discusses its
impact on noise and distortion. The impact of amplifier Gain Bandwidth Product on final
pole locations is also included, showing several examples. A complete high dynamic
range, differential I/O filter for wide dynamic range ADC driving is then designed. A
simple design spreadsheet embodying this design approach is available for download
withthisapplicationreport.
Contents
1 MFBFilterTransferFunction...................................................................... 2
2 MFBActiveFilterNoiseGainAnalysis.......................................................... 5
3 SettingtheIntegratorPoletoImproveNoiseandDistortion.................................. 7
4 ExampleDesignsShowingtheImpactofIntegratorPoleLocation.......................... 8
5 MFBFilterImplementedwithNon-Unity-GainStableOpAmps ........................... 12
6 DifferentialVersionof3rd-OrderDesign....................................................... 15
7 PoleSensitivitytoAmplifierGainBandwidthProduct........................................ 17
8 Summary........................................................................................... 20
9 References......................................................................................... 20
AppendixA OutputNoiseAnalysis .................................................................. 21
AppendixB SolutionforR andC .................................................................. 23
3 1
AppendixC EffectofanEqualCTarget ............................................................ 26
AppendixD NoiseGainZeroeswithC =¥ ........................................................ 27
1
AppendixE NoiseGainZeroesforR =R Targeted ............................................. 30
3 2
ListofFigures
1 MFBFilterTopology................................................................................ 2
2 DCAnalysisCircuit................................................................................. 3
3 FeedbackAnalysisCircuitforMFBFilter........................................................ 6
4 InitialTestCircuitusingtheOPA820ina1MHz,ButterworthLow-PassFilter
Configuration........................................................................................ 9
5 NoiseGainandOpen-LoopGainforCircuitinFigure4 ...................................... 9
6 NewDesignCircuitwithNoiseGainPeaking................................................. 10
7 NoiseGainPlotforFigure6..................................................................... 10
8 SimulatedSmallSignalBandwidthforFigure4andFigure6............................... 11
9 OutputSpotNoiseComparison................................................................. 12
10 NoiseGainPlotforInitialOPA2614MFBFilterDesign...................................... 13
11 NoiseGainwithOPA2614Open-LoopGainforC =31.8pF............................... 14
T
12 FrequencyResponsefortheOPA2614MFBFilterDesign(Each1/2ofFigure14)..... 14
13 ExpandedViewof3rd-OrderFilterResponse(seeFigure14)............................. 15
14 Single-SupplyDifferentialADCInterfacewith3rd-OrderBesselFilter(withf =
-3dB
FilterProisatrademarkofTexasInstruments.
Alltrademarksarethepropertyoftheirrespectiveowners.
SBOA114–February2006 DesignMethodologyforMFBFiltersinADCInterfaceApplications 1
SubmitDocumentationFeedback
www.ti.com
MFBFilterTransferFunction
5MHz)............................................................................................... 16
15 OPA2614Single+5VDistortionforNoninvertingDifferentialGainof8................... 17
A-1 NoiseAnalysisCircuitforMFBFilter........................................................... 21
1 MFB Filter Transfer Function
TheMultipleFeedBack(MFB)filteriswidelyusedforvery highdynamicrangeADC input stages.This
filteroffersexceptionalstopbandrejectionoverotherfilter topologies (Ref.1). Figure1showsthestarting
pointforthisfilterdesign.
R
1
C
2
R R
3 2
V
I
C1 VFB VO
Figure1. MFBFilter Topology
Thisdesignisaninvertingsignalpath,2nd-order, low-passfilter that offersnumerousadvantagesover
Sallen-Keyfilters.Thissingle-endedI/Ointerfacecanbeeasilyadapted toadifferentialI/O interface,as
willbeshownlater.
Afewoftheadvantagesofthistopologyinclude:
1. Nogainforthenoninvertingcurrentnoiseand/orDCbiascurrent.Figure1showsthenon-inverting
inputgrounded,whichisgreatforreducingnoisebut lessthanidealfor DCprecision. Addingaresistor
equalto theDCimpedancelookingouttheinvertingnodeachievesbias currentcancellation;adding a
capacitoracross thisresistorreducesthenoise contributionforthe resistorandthe opampbias
currentnoise.IftheamplifierisaJFETor CMOStype,thisbias current cancellationwillnotworkand
thenoninvertinginputshouldsimplybeset togroundor adesiredreferencevoltage.
2. Thein-bandsignalgainissetby–(R /R ). Aswillbeshown,R alsosetstheQ ofthe filterwhile
1 3 3
havingnoinfluenceoverw .
o
EmbeddedwithinthefilterisanIntegratorcomprisedofR andC , alongwiththeVoltageFeedBack
2 2
(VFB)opamp.Thisdesignnormallyneedsto beimplementedusingaunity-gainstable, VFBopamp
becausethecore gainelementneedstobeconfiguredasanIntegrator.Therearedynamicrange
advantagestousingnon-unity-gainstableVFBamplifiersandadesign approachforsuccessfullyapplying
thosetypes ofdeviceswillbeshownlater. However,aCurrent FeedBack(CFB) opampisusually not
suitableto thistypeoffiltersinceitslocalstability requiresafeedbackresistornearlyequalto a
recommendedvalue.AcapacitivefeedbackasrequiredinFigure1willtypicallynot workwithCFB amps
withoutsomedesigntricksthatusuallyimpairnoiseperformance (Ref.2). Sincethe emergingFully
DifferentialAmplifiers(FDA)areessentially voltagefeedback opamps,they canalsobeappliedquite
successfullytothistopology(Ref.3).
Numerousapproachestoselectingthecomponent valuesareavailable intheliterature(see, for example,
Ref.4).Anequal-Rapproachiscommon,andwillbeshown asadesirableapproachonceaninitialR is
2
chosen.AsADCscontinuetoimprove,the resistornoise inthisfiltercanactually beadominantelement
inthetotalnoisespectrumdeliveredtothe converterinput. Oneoutcomeofthe equal-Rdesign(Ref.4)is
thatquiteunequalCsarethenrequiredfor mostfilter targets.That isinfact generallytrueforthisfilter
type(aswillbeshownlater).IfanequalCdesignisdesired,anotherfilter typeshouldbeconsidered
(Sallen-Key).
2 DesignMethodologyforMFBFiltersinADCInterfaceApplications SBOA114–February2006
SubmitDocumentationFeedback
www.ti.com
MFBFilterTransferFunction
TheLaplacetransferfunctionforthecircuit ofFigure1isshownasEquation1.
Vo(cid:3) (cid:2)1 · 1
Vi C1C2R2R3 s2(cid:1)sC R1R (cid:4)R3(cid:1)R2(cid:4)1(cid:1)RR3(cid:5)(cid:5)(cid:1)R R1C C
1 2 3 1 1 2 1 2 (1)
Thevariousdesigngoalsofinterestinthisequationcanbesolvedas:
• theDCgain(wheres=0):
R
AV(cid:1)DC(cid:2)(cid:2)(cid:1)R1 (cid:4)inV(cid:3)V(cid:5)
3 (2)
• characteristicfrequency:
(cid:1)o(cid:1)(cid:2)R R1C C (cid:1)inradians(cid:2)
1 2 1 2 (3)
• thequalityfactor:
(cid:3)C
1
C
2
Q(cid:2) (cid:1)unitless(cid:2)
(cid:3)R (cid:3)R (cid:3)R R
1 2 1 2
(cid:1) (cid:1)
R R R
2 1 3 (4)
R
Or, with(cid:1)(cid:3) 1
R
2,
(cid:1)(cid:1)(cid:1)
(cid:6)C
R 1
3 C
(cid:1)(cid:1)(cid:1)Q(cid:2) 2
R2(cid:6)(cid:1)(cid:1)R3(cid:4)(cid:6)(cid:1)(cid:1)(cid:6)1(cid:1)(cid:5) (5)
Asisusuallythecaseinactivefilterdesign, therearemorepassive elementstoberesolvedthanfilter
characteristics.Here,therearefiveelementsandonlythreetargets. Thissituationoftenleads tothevery
commonequal-Rassumptiontoreachadesign wherethat isasomewhatarbitraryway toeliminateone
degreeoffreedom.Thereshouldbeamore rationalway toselectcomponent valuesfor thesefilters.
Lookingat thecircuit ofFigure1atDCgivesthesimplifiedcircuitof Figure2thatwillbeusedtoshowthe
DCpartofthelow-passfilter.
R
1
R R
3 2
V
I
VO
R
P
Set:R =R +R œœR
P 2 1 3
Figure2. DCAnalysisCircuit
Aresistor(R )hasbeenaddedonthenoninvertinginput toprovidefor DCbiascurrentcancellationin the
P
outputoffsetvoltage.Settingitasshownreducestheoutput DCerrorto(I •R )ifthe opampshowsan
OS 1
inputbiascurrentthathasanoffsetcurrentspecification. Again, JFETorCMOSamplifierswouldnot use
thisR resistorforoutputDCerrorreduction.
P
SBOA114–February2006 DesignMethodologyforMFBFiltersinADCInterfaceApplications 3
SubmitDocumentationFeedback
www.ti.com
MFBFilterTransferFunction
Thetotaloutputnoiseisaconsiderablymoreinvolveddiscussion. AppendixAdevelops that expression
forthecircuit ofFigure2,wherethebandlimitingeffectsofthe filtercapacitorsareneglected. Thetotal
outputnoiseisgivenasEquation6(whichisalsoEquationA-3inAppendixA), wherethetermsarising
fromR areneglected.
P
(cid:7) 2 2 2
e (e )2(cid:5)1 R1(cid:6) 4kTR (cid:5)1 R1(cid:6) 4kTR (cid:5)1 R1(cid:6) i 2(cid:3)R (cid:5)1 R1(cid:6) R (cid:4)
o(cid:2) n (cid:1)R (cid:1) 2 (cid:1)R (cid:1) 1 (cid:1)R (cid:1)n 2 (cid:1)R (cid:1) 1
3 3 3 3
(6)
Itwouldbereasonabletoassumethatmost designswouldnotwant the resistortermstoaddtoomuch
morenoiseattheoutputthantheopampnoisevoltageitself.Thisidea canbeusedto develop
Equation7(EquationA-4inAppendixA)wheretheopampnoise voltage(squared)istargetedtobe
equaltothetotalnoisepowercontributionofthe R andR resistorsat theoutput.
2 3
2 2
(cid:3)1(cid:1)RR1(cid:4) en2(cid:2)(cid:3)4kTR2(cid:1)(cid:3)inR2(cid:4)2(cid:4)(cid:3)1(cid:1)RR1(cid:4) (cid:1)in2(cid:3)2R2R1(cid:3)1(cid:1)RR1(cid:4)(cid:4)
3 3 3
(7)
ThisexpressionstillincludesbothR andR . R isalso aresistorthat willcontributetothe totaloutput
1 3 3
noiseinasimilarfashiontoR .Itwouldbepreferableto makeitaslowaspossiblewithinthe constraint
2
thatitshouldnotloadthedrivingsignalsourcetothe pointof creatingadominantdistortionmechanismin
thatpriorstage.As amaximumvalue, itmightbereasonableto letitequalR whilerecognizingthat
2
movingitlowerwillbenefitthetotaloutput noise. If R istentatively setequalto R , Equation7can be
3 2
simplifiedandputintoaformtosolveforR , asshowninEquation8(fromEquationA-13inAppendixA).
2
2
R22(cid:1)R2(cid:4)11(cid:1)(cid:1)3AAVV(cid:5) (4ikn2T)(cid:2)(cid:4)11(cid:1)(cid:1)3AAVV(cid:5)(cid:4)einn(cid:5) (cid:3)0 (8)
Thisequationmaythenbesolvedusingthe quadraticequationforaninitialtargetvaluefor R asshown
2
inEquation9(whereonlythepositivesolutionforR isused;notethat Equation9isalso Equation A-14in
2
AppendixA).
R2(cid:3)(cid:4)11(cid:1)(cid:1)3AAVV(cid:5) (2ink2T)(cid:7)(cid:8)(cid:9)(cid:6)1(cid:1)(cid:4)11(cid:1)(cid:1)3AAVV(cid:5)(cid:4)2eknT(cid:5)2 (cid:2)1 (cid:7)(cid:10)(cid:11)
(9)
Thisformulagivesaninitialsuggestedvaluefor R (notethat embeddedinthissolution istheassumption
2
thatR willthenbesetequaltoR ).BothR andR maybeset lowertoimprovenoise.Recognize,
3 2 2 3
however,thatverylowvalueswillstarttoload theoutput stagedriving intothisfilter andthefilterop amp
outputstage(ifR isalsoverylow).Theycanalso besethigher tolightenthe loadingwhereanincrease
1
intotaloutputnoisewillbetheresult.
OnceR isselected,eitherfromthisnoiseconsiderationor fromsomeotherapproach, wenowneedto
2
selectoneofthecapacitorvaluestoremoveitfromthefilter designequations.Withtwoelements
selected,theremainingthree canbeusedto setthe threefilter designgoals. Insolvingto achieve the
desiredfiltershape,itispossible(AppendixB)to arriveatanequationfor C that showsacritical
1
constraintontheR C product.Thatconstraint canbeseeninEquation10(which istaken from
2 2
AppendixBasEquationB-22):
Q
C1(cid:3)(cid:1)oR2(cid:4)1(cid:2)Q(cid:1)oR2C2(cid:1)1(cid:1)AV(cid:2)(cid:5) (10)
Equation10clearlyshowsthattheR C productmust belowenoughtokeepthe solutionfor C >0. That
2 2 1
constraintisshownasEquation11:
(cid:1) R C (cid:2) 1 for C (cid:3)0
o 2 2 Q(cid:1)1(cid:1)A (cid:2) 1
V (11)
or:
1 (cid:2)Q(cid:1)1(cid:1)A (cid:2) (cid:3)ratioofIntegratorpoletotarget(cid:1) (cid:4)
(cid:1) R C V o
o 2 2 (12)
4 DesignMethodologyforMFBFiltersinADCInterfaceApplications SBOA114–February2006
SubmitDocumentationFeedback
www.ti.com
MFBActiveFilterNoiseGainAnalysis
(1/w R C )isphysicallytheratiooftheIntegratorpoleto thetargetw . Equation12sets aminimumlimit
o 2 2 o
forthatratio,whilemovingabovethatlimitisessentiallymovingtheIntegratorpoleout relativetow
o
(reducingC ifw andR arefixed). Solvingforequality inEquation12solvesfor aninfinitevalue of C .
2 o 2 1
So,movingtheIntegratorpoleoutalsoreducesthe requiredvaluefor C frominfinityto somemore
1
reasonablevalue.
WhileEquation11andEquation12giveanicelimit onasolutionfor C andC (notethat onceC is
2 1 2
selected,Equation10givesC completelydefinedbythe desiredfiltershapeandaselectedR value),
1 2
theydonottellusmuch aboutwhere toplacethe Integratorpole.
Oneoptionconsidered formanyactive filterdesignsistoset thetwocapacitorsequal. Thissolutionis
sometimesconsideredpreferabletogetbettermatchinginorderto minimizethesensitivity functions.
Equation10actuallygivesusaneasywayto testthisapproachbytemporarily targetingC = C ,then
1 2
solvingtheresultingexpressionforC (AppendixC). Equation13showstheresultingquadraticwhile
2
Equation14showsthesolutionforC .Again,thissimplifiedapproachisonly possibleifR isinitially
2 2
selectedfromeitheranoiseapproachor someotherconsideration.
C2 (cid:2) C 1 (cid:1) 1 (cid:3)0
2 2 R2Q(cid:1)o(cid:1)1(cid:1)AV(cid:2) (R2(cid:1)o)2(cid:1)1(cid:1)AV(cid:2) (13)
(whichisEquationC-3inAppendixC;)
C2(cid:4)2QR2(cid:1)1o(cid:1)1(cid:2)AV(cid:2)(cid:5)1(cid:1)(cid:7)1(cid:3)(2Q)2(1(cid:2)AV)(cid:6) (14)
(whichisEquationC-10inAppendixC;)
TheradicalinEquation14onlysolvesfor non-imaginaryC valuesif(2Q)2•(1 +A )£ 1.Assuming a
2 V
minimumA =1requiresaQ<0.353.SettingQ= 0.353andA =1givesaC solutionthat istwo
V V 2
repeatedvaluesgiveninEquation15:
C (cid:1)C (cid:1) 1
2 1 1.43R (cid:1)
2 o (15)
UsinganactivefilterforaQ<0.5(tworealpoles) and/orgain <1isunlikelybecauseasimplepassive
circuitcaneasilyprovide attenuationandtworealpoleswithout requiringanactiveelement. Thus,an
equalC designis interestingbutnotparticularlyusefulfor anMFBfilter design.
2 MFB Active Filter Noise Gain Analysis
Onepossibleapproachtosettingthe(1/R C )product (Integratorpolelocation) isto investigatewhat
2 2
impactitmighthaveontheresultingnoisegainfor thecompletedfilter. Recallthat the noisegain isthe
reciprocalofthefeedbackattenuationfromthe opampoutput pinbacktothe invertinginput pin. This
noisegainisanimportantconsideration forat least threereasons.
1. Anypeakinginthenoisegainwill, of course,peakupthe gaintothe outputfor the totalequivalent
inputvoltagenoise(includingtheR effectsconsideredearlier).
2
2. Noisegainpeakingwill alsoreducethe loopgain. Allotherthingsbeing equal, reducedloopgainwill
showupashigheroutputharmonicdistortion.
3. Thepointwhere thenoisegaincrossestheopen-loopgain(inaBodeplot) alsodeterminestheloop
gainphasemargin.Phase marginat thiscrossoverpoint setsthe stabilityfor theoveralldesign.
Thefeedbackdividercircuit fortheMFBfilterisshowninFigure3. Anaddedelementisincludedherethat
wasnotinFigure1—aparasiticorintentionalcapacitor(C )ontheinvertinginput pin.
T
SBOA114–February2006 DesignMethodologyforMFBFiltersinADCInterfaceApplications 5
SubmitDocumentationFeedback
www.ti.com
MFBActiveFilterNoiseGainAnalysis
R
1
C2 V! R2
V
O
C C R
T 1 3
Source input, assumed
low impedance.
Figure3.FeedbackAnalysisCircuit forMFBFilter
Here,itiseasytoseethemultiplefeedbacknatureof thecircuit.AtDC, thepath isthroughR and R ,
1 2
whileathighfrequenciesitisthroughC .
2
ThedesiredLaplacetransferfunctionhereisfromV to V–(theinvertinginput voltage). Thisattenuatoris
O
normallyreferredtoastheb incontroltheorydiscussionsofnegativefeedbacksystems.Thesolutionfor
b isshownasEquation16.
s2 s(cid:6) 1 1 (cid:7) 1
(cid:1) (cid:1) (cid:1)
(cid:1)(cid:4)V(cid:2)(cid:3) C2 · C1R3 C1(cid:8)R1(cid:5)R2(cid:9) R1R2C1C2
V C C
O T(cid:1) 2 s2 s(cid:6) 1 1 1 (cid:7) 1
(cid:1) (cid:1) (cid:1) (cid:1)
C1R3 C1(cid:8)R1(cid:5)R2(cid:9) R2(cid:8)CT(cid:1)C2(cid:9) R2(cid:8)R1(cid:5)R3(cid:9)C1(cid:8)CT(cid:1)C2(cid:9) (16)
Wearenormallymoreinterestedinlookingat 1/b , whichwillbethenoisegaindiscussedearlier.
InvertingEquation16givesEquation 17.
s2 s(cid:4) 1 1 1 (cid:5) 1
(cid:1) (cid:1) (cid:1) (cid:1)
1 (cid:6)1 CT(cid:7)· C1R3 C1(cid:6)R1(cid:3)R2(cid:7) R2(cid:6)CT(cid:1)C2(cid:7) R2(cid:6)R1(cid:3)R3(cid:7)C1(cid:6)CT(cid:1)C2(cid:7)
(cid:2) (cid:1)
(cid:1) C
2 s2 s(cid:4) 1 1 (cid:5) 1
(cid:1) (cid:1) (cid:1)
C1R3 C1(cid:6)R1(cid:3)R2(cid:7) R1R2C1C2 (17)
Thereareseveral keypointstoEquation17:
1. Thepolesare identicaltothedesiredfilterpoles.
2. TheDC(s=0)gainbecomes(1+R /R ).
1 3
3. Thehighfrequencygain(assfi ¥ )goesto(1+ C /C ).
T 2
4. IfC <<C ,thenthehighfrequencynoisegain approaches1.
T 2
Thenoisegaintransferfunctionhastwozeroesandtwopoles. ThetransitionbetweentheDCgainand
highfrequencygaindependsontherelativepositionof thezeroesandpoles. It wouldbepreferable to
minimizethepeakinginthenoisegainwithinthe desiredlow-pass frequencybandasitmakesthis
transitionfromtheDCgaintothesfi ¥ gain. If the desiredfilter shapecallsfor ahighQ, peakingin this
noisegainresponseisunavoidableasthefrequencyapproachesw . If,however, oneor bothzeroesare
o
placedtofall belowthew frequency,addednoisegain peakingresultsthat mightbeunnecessary.The
o
questiontheniswhetherornotthosezeroescanbeplacedabovew inorder tolimitanyadditional
o
in-bandpeakingforthenoisegain.
StartingfromEquation17,firstsetC =0for thisportionof theanalysis.It willbeusedlaterto allow
T
non-unity-gainstableamplifierstobeusedintheMFBtopology,but willunnecessarilycomplicatethe
resolutionof(1/R C ).RewritingEquation17withthissimplificationyieldsEquation18.
2 2
s2 s(cid:4) 1 1 1 (cid:5) 1
(cid:1) (cid:1) (cid:1) (cid:1)
1 C1R3 C1(cid:6)R1(cid:3)R2(cid:7) R2C2 R2(cid:6)R1(cid:3)R3(cid:7)C1C2
(cid:2)
(cid:1)
s2 s(cid:4) 1 1 (cid:5) 1
(cid:1) (cid:1) (cid:1)
C1R3 C1(cid:6)R1(cid:3)R2(cid:7) R1R2C1C2 (18)
6 DesignMethodologyforMFBFiltersinADCInterfaceApplications SBOA114–February2006
SubmitDocumentationFeedback
www.ti.com
SettingtheIntegratorPoletoImproveNoiseandDistortion
Then,observingthatthenumeratorcoefficientsare verynearlythesameasthe denominatorcoefficients,
wecanrewritethenoisegainequationintermsofthedesired filter characteristicsasshownin
Equation19.
s2(cid:1)s(cid:3)(cid:2)Qo(cid:1)R1C (cid:4)(cid:1)(cid:1)1(cid:1)AV(cid:2)(cid:2)o2
1 2 2
(cid:2) (cid:3)MFBnoisegain(cid:4)
(cid:1) s2(cid:1)s(cid:3)(cid:2)Qo(cid:4)(cid:1)(cid:2)o2 (19)
Here,itbecomesveryapparentthattheembeddedIntegratorpole location,(1 /R C ), isthe oneadded
2 2
degreeoffreedominsettingthenoisegainzeroes. Allofthe otherelementsfeedingintothe noisegain
zerocoefficientsaresetbythedesiredlow-pass, 2nd-orderfilter terms.It isthereforeveryusefulto cast
thedesignanalysisintermsofthisIntegratorpole location.Morespecifically,workingintermsoftheratio
(1/w R C )allowsbettersimplificationandissimply theratioof thetargetcharacteristic frequencyand
o 2 2
theembeddedIntegratorpolelocation.
3 Setting the Integrator Pole to Improve Noise and Distortion
Onelimittotherangeon(1/w R C )hasalreadybeensetinEquation12to achieverealvaluesforC .
o 2 2 1
Setting(1 /w R C )equaltoQ(1+A )willsolveC for infinityandR for zero.Neither of these values
o 2 2 V 1 3
areparticularlyusefulforimplementation, butareinterestingasalimitto thenoise gainzerolocations.
UsingEquation12tosetaminimumlevelfor1/ R C givesw •Q•(1+A ). Puttingthisresultintothe
2 2 o V
numeratorofEquation19,andsolvingforthe equivalentQ for the zeroesofthe noisegainintermsof
C
thedesiredfiltershapeterms,givesEquation20(from AppendixD,EquationD-9).
QC(cid:2)1(cid:1)QQ(cid:3)21(cid:1)(cid:1)1(cid:1)AAVV(cid:2)(cid:2)1(cid:1)xx2 [where x(cid:2)Q(cid:3)1(cid:1)AV ] (20)
Thisresultisveryusefulinthatitclearlyshowsthe maximumpossiblevaluefor Q isone-half. (1)This
C
occursforanycombinationofdesiredA andQthat sets Q•(cid:214) (1+ Av)= 1.Themost commoncondition
V
forthiswouldbeQ=0.707(Butterworthtarget) andA = 2.Thatcombination,withatargetfor the
V
IntegratorpolesetbyEquation12,givesrepeatedrealzeroesatw /Q (AppendixD). Severalimportant
o
conclusionscomefromthiscombination.
1. Thenoisegainzeroesarealwaystworealzeroes.
2. Theyarerepeated,andatthemaximum value, atw / Qfor thespecific conditionsdescribedabove
o
(whichis notrealizablesinceC =¥ ).
1
3. ForQ>1,itwillnecessarilybethecasethat oneofthezeroes willfallbelowthe targetw . This
o
observationsaysthathigherQtargetsintheMFBfilterhaveanaddedpeakinginthe noisegain
beyondjustthedesiredfilterpolepeakingbecauseoneof thenoisegain zeroes movesbeloww .
o
4. Movingthetarget(1/w R C )up(movingtheIntegratorpoleout) to getarealsolutionfor C
o 2 2 1
effectivelymovesoneofthezeroesupinfrequencyandthe otherdown infrequencyalongthe
negativerealaxisinthes-plane.
5. Itturnsoutthatthishigherzerolocationcorrespondsvery closelyto1/ R C asthe zeroesbecome
2 2
widelyseparated.
Asthe(1/w R C )targetisincreasedfromitsminimumof Q•(1+ A ),C comesdownfrominfinityand
o 2 2 V 1
R increasesfromzero.Thesechangesarebothdesirablewithinsome limits.AsR startstoincrease
3 3
beyondthetargetedR value(fromthenoiseanalysisofEquation9), itwillalsostartto addmeaningfully
2
tothetotaloutputnoise.Onepossiblelimit toR isto setitequaltoR andresolvewhatthismeansfora
3 2
(1/w R C )target.Equation21showsthisresult(from AppendixE).
o 2 2
1 (cid:2)Q(cid:1)2A (cid:1)1(cid:2) [where R (cid:2)R isdesired]
(cid:1) R C V 3 2
o 2 2 (21)
AppendixEgoesontosolvefortheresultingzerolocationifEquation21isusedto set1/ R C .That
2 2
resultisshownasEquation22(EquationE-16fromAppendixE).
(cid:8) (cid:7) 2(cid:10)
Z1,2(cid:4)(cid:3)2(cid:1)Qo (cid:5)1(cid:2)Q2(cid:5)2AV(cid:2)1(cid:6)(cid:6)(cid:12)1(cid:1) 1(cid:3)(cid:5)1(cid:2)2QQ2(cid:7)(12(cid:2)AVA(cid:2)V1)(cid:6)(cid:12)
(cid:9) (cid:11)
(22)
(1) x/(1+x2)reachesamaximumvalueequalto1/2over0£ x£ ¥ atx=1.
SBOA114–February2006 DesignMethodologyforMFBFiltersinADCInterfaceApplications 7
SubmitDocumentationFeedback
www.ti.com
ExampleDesignsShowingtheImpactofIntegratorPoleLocation
Setting1/R C asgiveninEquation 21getsR =R . It alsosetsthe twonoisegain zeroesasgiven by
2 2 3 2
Equation22.Movingthetarget(1/w R C ) belowthevalueset byEquation21pullstheIntegratorpole
o 2 2
down,movesthelowerzerotoahigherfrequency(actingtoreducethe noisegainpeaking), andreduces
R belowR .Allofthesedesirableeffectssuggest that (1/w R C ) shouldbesetslightlybelowthevalue
3 2 o 2 2
determinedbyEquation21aslongasthe resultingR isnotsolowastopresent tooheavyaloadforthe
3
drivingsourceintothefilter.SinceR inFigure 1looks intoavirtualgroundnodein-band,the input
3
impedancebelowcutoffwillbeR .Conversely,movingthe(1 /w R C )abovethevaluesetby
3 o 2 2
Equation21increasesR beyondthetargetedR value,movesthe lowernoisegain zerodownfurther
3 2
(causingaddedpeakinginthenoisegainwithinthe desiredfrequencypassband), andextends the higher
noisegainzerooutinfrequency(approximatingthe1/R C Integratorpolelocation).
2 2
Insummary,the(1/w R C )targetshouldbeset withintherangeindicatedinEquation23.
o 2 2
Q(cid:1)A (cid:1)1(cid:2) (cid:2) 1 (cid:3) Q(cid:1)2A (cid:1)1(cid:2)
V (cid:1) R C V
(cid:3)(cid:3) o 2 2 (23)
Setting 1 (cid:2)(cid:1) Q(cid:1)2A (cid:1)1(cid:2) gives R (cid:2)R
R C o V 3 2
2 2
(cid:3)(cid:3)
whilesetting 1 (cid:2)(cid:1) Q(cid:1)A (cid:1)1(cid:2) gives C (cid:4)(cid:3) and R (cid:2)0
R C o V 1 3
2 2 (24)
4 Example Designs Showing the Impact of Integrator Pole Location
Nowlet'susetheseresultstostepthroughadesign andobservethenoise anddistortion thatresultsfrom
variousselectionsof(1/w R C ).Allof thesedesigns givethe desiredfiltershape. It isthe noise gain
o 2 2
shapeandnoisecontributionsoftheresistorvalues that areofinterest here.
Startwithatargetdesigngivenby:
• w =2p •1MHz
o
• Q=0.707
• A =2(gives anegativegainof2for thesignalpath)
V
Then,pickanamplifiertogetitsnoiseandopen-loopgaincharacteristic.Forthisfirst example,wewill
usethesinglechannelOPA820—arelativelylow-noise,unity-gainstable, widebandVFB opamp.The
necessaryspecificationsforthispart ofthe designareshown inTable1.
Table1.OPA820NoiseandOpen-LoopGain
Specifications
PARTNO. GBW(MHz) e (nV/(cid:214) Hz) A (V/V) i (pA/(cid:214) Hz)
n OL n
OPA820 280 2.5 2000 1.7
ThetargetedmaximumR valueiscalculatedusing Equation9, givingthisresult:
2
• SuggestedR =366.38W
2
PickingR =250W willallowustoproceedtosetting the(1/w R C )target. Therecommendedrange
2 o 2 2
(usingEquation23)is:
• MinimumallowedratioofIntegrator/w pole is:2.12
o
– Thisresult setsalimittogettingavalidsolutionforC
1
• MaximumvaluetogetR =R is:3.53
3 2
– Thisresult setsR =R fornoise control
3 2
IftheR =R valueischosen,theresultingzero locationsare(from Equation22):
3 2
• First,computetheradicalforthepolynomial:0.714241334
– Thelowerzerolocationis707kHz
– Theupperzero locationis4.24MHz
8 DesignMethodologyforMFBFiltersinADCInterfaceApplications SBOA114–February2006
SubmitDocumentationFeedback
www.ti.com
ExampleDesignsShowingtheImpactofIntegratorPoleLocation
ContinuingwiththeR =R solution,thefinalcomponent valuesto placeintoFigure 1andthe design
3 2
sequenceare:
• R =250W [selectedtocontrolnoiseusingEquation9]
2
• C =180pF[setbytargeting(1/w R C ) =Q(2A +1)]
2 o 2 2 V
• C =1130pF[setusingEquation10]
1
• R =250W [setusingEquationB-17]
3
• R =500W [setusingEquation2]
1
• DCGain(A )=2.00V/V [fromEquation2]
V
• F =1MHz=F (ifQ=0.707)[fromEquation3]
O –3dB
• Q=0.707[fromEquation4]
ThenoisegainandphasecanbecomputedusingEquation17whereC =3pFisusedto emulatea
T
parasiticcapacitanceontheinvertingopamp input.Thisnoise gainandphasecanbeplotted alongwith
theopen-loopgainandphasefortheOPA820.Finally, the phasemarginwherethisnoisegain intersects
theopen-loopresponsecanbederived.
TheexamplecircuitdevelopedhereisshownasFigure4.
500W
180pF
+5V
250W 250W
V
I
1130pF OPA820 V
O
100W
0.1mF 417W
-5V
Figure4.InitialTestCircuitusingtheOPA820in a1MHz,ButterworthLow-PassFilterConfiguration
ThiscircuitgivestheloopgainplotofFigure5.
MFB OVERALL NOISE GAIN
90
OPA820 Open-Loop Gain
60
Noise-Gain Magnitude
(cid:176)e () 30
as 0
h
d P -30 Noise-Gain Phase
n
B) a -60
d
n ( -90
ai
G -120
-150 OPA820 Open-Loop Phase
-180
3 4 5 6 7 8 9
10 10 10 10 10 10 10
Frequency (Hz)
Figure5.NoiseGain andOpen-LoopGainforCircuitinFigure4
Thisplottransitionsfairlysmoothlyfromanoisegainof20log(3)= 9.5dBto0dBwithonly minorpeaking
asaresultofthenoisegainzeroat707kHz.Phasemarginfor thiscircuitis51degrees,whichisquite
stable.
SBOA114–February2006 DesignMethodologyforMFBFiltersinADCInterfaceApplications 9
SubmitDocumentationFeedback
www.ti.com
ExampleDesignsShowingtheImpactofIntegratorPoleLocation
Inordertotesttheimpact ofsettingthe(1/w R C )targetfar toohigh, set itto10andrepeatthedesign
o 2 2
inthismanner(stillholdingR =250W ):
2
• R =250W [selectedtocontrolnoiseusingEquation9]
2
• C =63.7pF[setbytargeting(1/w R C ) =10]
2 o 2 2
• C =571pF[setusingEquation10]
1
• R =1.39kW [setusingEquationB-17]
3
• R =2.79kW [setusingEquation2]
1
• DCGain(A )=2.00V/V[fromEquation2]
V
• F =1MHz[fromEquation3]
O
• Q=0.707[fromEquation4]
Figure6showsthenewdesigncircuit withthese morewidelyseparatednoise gainzeroes(thesezeroes
arenowat268kHzand10.7MHz,solvingthenumeratorof Equation19).
2.79kW
64pF
+5V
1.39kW 250W
V
I
571pF OPA820 V
O
100W
0.1mF 1.18kW
-5V
Figure6.NewDesignCircuitwithNoiseGainPeaking
MFB OVERALL NOISE GAIN
90
OPA820 Open-Loop Gain
60
Noise-Gain Magnitude
(cid:176)e () 30
as 0
h
d P -30
n
B) a -60 Noise-Gain Phase
d
n ( -90
ai
G -120
-150 OPA820 Open-Loop Phase
-180
3 4 5 6 7 8 9
10 10 10 10 10 10 10
Frequency (Hz)
Figure7. NoiseGainPlot forFigure6
Theloopphasemarginisnotimpactedgreatly, goingto54degrees. Thisvalueisslightlybetterthan the
previousone,andstillverystable.Theplot of Figure7clearlyshowsthislower zerointhe phase
response.Thenoisegainphasecurvepeaksupmuchmoreasthezerocomesinearlierandbefore the
twopolesreverseit.Moreimportantly,thenoisegain peaksupslightly, reducingthe availableloop gain in
thepassband.
10 DesignMethodologyforMFBFiltersinADCInterfaceApplications SBOA114–February2006
SubmitDocumentationFeedback