Design Methodology for MFB Filters in ADC Interface ...

1 ApplicationReportSBOA114 (MFB)filtertopologyiswell-known,itsappli cationtoveryhighdynamicrangeanalog-to-di gitalconverter(ADC) , , , ..27 AppendixENoiseGainZeroesforR3= , (Each1/2ofFigure14)..1413 ExpandedViewof3rd-OrderFilterResponse(se eFigure14)..1514 Single-SupplyDifferentialADCI nterfacewith3rd-OrderBesselFilter(withf- 3dB= )..1615 OPA2614 Single+ (MFB) ( ). ,2nd-order, , , ; , (R1/R3).Aswillbeshown,R3alsosetstheQofth efilterwhilehavingnoinfluenceover ,alongwiththeVoltageFeedBack(VFB) , ,aCurrentFeedBack(CFB) ( ).SincetheemergingFullyDifferentialAmpli fiers(FDA)areessentiallyvoltagefeedbacko pamps,theycanalsobeappliedquitesuccessfu llytothistopology( ).Numerousapproachestoselectingthecompon entvaluesareavailableintheliterature(see ,forexample, ).Anequal-Rapproachiscommon, , ( ) (aswillbeshownlater).IfanequalCdesignisd esired,anotherfiltertypeshouldbeconsider ed(Sallen-Key).

2 2 DesignMethodologyforMFBF iltersinADCI nterfaceApplicationsSBOA114 1C1C2R2R3 1s2 s1C1R2R3 R3 R2 1 R3R1 1R1R2C1C2(1)AV DC R1R3 in V V (2) o 1R1R2C1C2 in radians (3)Q C1C2 R1R2 R2R1 R1R2 R3 unitless (4)Or, with R1R2, Q R3C1C2 R2 R3 1 (5)RPSet:=+RRRRP213 : theDCgain(wheres=0): characteristicfrequency: thequalityfactor:Asisusuallythecaseinact ivefilterdesign, , (RP) (IOS R1) , (en)2 1 R1R3 2 4kTR2 1 R1R3 2 4kTR1 1 R1R3 in2 R2 1 R1R3 R1 2 (6) 1 R1R3 2en2 4kTR2 inR2 2 1 R1R3 2 in2 2R2R1 1 R1R3 (7)R22 R2 1 AV1 3AV 4kT(in2) 1 AV1 3AV enin 2 0(8)R2 1 AV1 3AV 2kT(in2) 1 1 3AV1 AV en2kT 2 1 (9)C1 Q oR2 1 Q oR2C2 1 AV (10) oR2C2 1Q 1 AV forC1 0(11)1 oR2C2 Q 1 AV ratio of Integrator pole totarget o (12) , (whichisalsoEquationA-3inAppendixA), (EquationA-4inAppendixA)wheretheopampnoi sevoltage(squared) , ,Equation7canbesimplifiedandputintoaform tosolveforR2,asshowninEquation8(fromEqua tionA-13inAppendixA).

3 Thisequationmaythenbesolvedusingthequadr aticequationforaninitialtargetvalueforR2 asshowninEquation9(whereonlythepositives olutionforR2isused;notethatEquation9isal soEquationA-14inAppendixA).Thisformulagi vesaninitialsuggestedvalueforR2(notethat embeddedinthissolutionistheassumptiontha tR3willthenbesetequaltoR2). ,however,thatverylowvalueswillstarttoloa dtheoutputstagedrivingintothisfilterandt hefilteropampoutputstage(ifR1isalsoveryl ow). ,eitherfromthisnoiseconsiderationorfroms omeotherapproach, , ,itispossible(AppendixB) (whichistakenfromAppendixBasEquationB-22 ):Equation10clearlyshowsthattheR2C2produ ctmustbelowenoughtokeepthesolutionforC1> :or:4 DesignMethodologyforMFBF iltersinADCI nterfaceApplicationsSBOA114 C21R2Q o 1 AV 1(R2 o)2 1 AV 0(13)C2 12 Q R2 o 1 AV 1 1 (2Q)2(1 AV) (14)C2 C1 o(15)2 MFBA ctiveFilterNoiseGainAnalysisMFBA ctiveFilterNoiseGainAnalysis(1/ oR2C2)isphysicallytheratiooftheIntegrato rpoletothetarget ,whilemovingabovethatlimitisessentiallym ovingtheIntegratorpoleoutrelativeto o(reducingC2if oandR2arefixed).

4 , (notethatonceC2isselected,Equation10give sC1completelydefinedbythedesiredfiltersh apeandaselectedR2value), ,thensolvingtheresultingexpressionforC2( AppendixC). ,thissimplifiedapproachisonlypossibleifR 2isinitiallyselectedfromeitheranoiseappr oachorsomeotherconsideration.(whichisEqu ationC-3inAppendixC;)(whichisEquationC-1 0inAppendixC;)TheradicalinEquation14only solvesfornon-imaginaryC2valuesif(2Q)2 (1+AV) < :UsinganactivefilterforaQ< (tworealpoles)and/orgain< , (1/R2C2)product(Integratorpolelocation) ,ofcourse,peakupthegaintotheoutputforthe totalequivalentinputvoltagenoise(includi ngtheR2effectsconsideredearlier). , (inaBodeplot) aparasiticorintentionalcapacitor(CT) !C2 VOR1 CTC1R3 Source input, assumedlow impedance. V VO C2CT C2 s2 s 1C1R3 1C1 R1 R2 1R1R2C1C2s2 s 1C1R3 1C1 R1 R2 1R2 CT C2 1R2 R1 R3 C1 CT C2 (16)1 1 CTC2 s2 s 1C1R3 1C1 R1 R2 1R2 CT C2 1R2 R1 R3 C1 CT C2 s2 s 1C1R3 1C1 R1 R2 1R1R2C1C2(17)1 s2 s 1C1R3 1C1 R1 R2 1R2C2 1R2 R1 R3 C1C2s2 s 1C1R3 1C1 R1 R2 1R1R2C1C2(18) , ,thepathisthroughR1andR2, (theinvertinginputvoltage).

5 Thisattenuatorisnormallyreferredtoasthe , (s=0)gainbecomes(1+R1/R3). (ass )goesto(1+CT/C2). <<C2, ,peakinginthisnoisegainresponseisunavoid ableasthefrequencyapproaches ,however,oneorbothzeroesareplacedtofallb elowthe ofrequency, ,firstsetCT= ,butwillunnecessarilycomplicatetheresolu tionof(1/R2C2). s2 s oQ 1R2C2 1 AV o2s2 s oQ o2 MFB noise gain (19)3 SettingtheIntegratorPoletoImproveNoisean dDistortionQC Q1 AV 1 Q2 1 AV x1 x2[ wherex Q1 AV ](20)1 oR2C2 Q 2AV 1 [ whereR3 R2is desired ](21)Z1,2 o2Q 1 Q2 2AV 1 1 1 2Q1 AV 1 Q2(2AV 1) 2 (22)SettingtheIntegratorPoletoImproveNoi seandDistortionThen,observingthatthenume ratorcoefficientsareverynearlythesameast hedenominatorcoefficients, ,itbecomesveryapparentthattheembeddedInt egratorpolelocation,(1/R2C2), , ,workingintermsoftheratio(1/ oR2C2) (1/ oR2C2) (1/ oR2C2)equaltoQ(1+AV) , o Q (1+AV).

6 PuttingthisresultintothenumeratorofEquat ion19,andsolvingfortheequivalentQCforthe zeroesofthenoisegainintermsofthedesiredf iltershapeterms,givesEquation20(fromAppe ndixD,EquationD-9).Thisresultisveryusefu linthatitclearlyshowsthemaximumpossiblev alueforQCisone-half.(1)Thisoccursforanyc ombinationofdesiredAVandQthatsetsQ (1+Av)= (Butterworthtarget)andAV= ,withatargetfortheIntegratorpolesetbyEqu ation12,givesrepeatedrealzeroesat o/Q(AppendixD). ,andatthemaximumvalue,at o/Qforthespecificconditionsdescribedabov e(whichisnotrealizablesinceC1= ). >1,itwillnecessarilybethecasethatoneofth ezeroeswillfallbelowthetarget (1/ oR2C2)up(movingtheIntegratorpoleout) (1/ oR2C2)targetisincreasedfromitsminimumofQ (1+AV), (fromthenoiseanalysisofEquation9), (1/ oR2C2) (fromAppendixE).AppendixEgoesontosolvefo rtheresultingzerolocationifEquation21isu sedtoset1 (EquationE-16fromAppendixE).

7 (1)x/(1+x2)reachesamaximumvalueequalto1/ 2over0 x atx= AV 1 1 oR2C2 Q 2AV 1 (23) Setting1R2C2 oQ 2AV 1 gives R3 R2 while setting1R2C2 oQ AV 1 gives C1 andR3 0(24)4 ExampleDesignsShowingtheImpactofIntegrat orPoleLocationExampleDesignsShowingtheIm pactofIntegratorPoleLocationSetting1/R2C 2asgiveninEquation21getsR3= (1/ oR2C2)belowthevaluesetbyEquation21pullst heIntegratorpoledown,movesthelowerzeroto ahigherfrequency(actingtoreducethenoiseg ainpeaking), (1/ oR2C2) , ,movingthe(1/ oR2C2)abovethevaluesetbyEquation21increa sesR3beyondthetargetedR2value,movesthelo wernoisegainzerodownfurther(causingadded peakinginthenoisegainwithinthedesiredfre quencypassband),andextendsthehighernoise gainzerooutinfrequency(approximatingthe1 /R2C2 Integratorpolelocation).Insummary,the(1/ oR2C2) 'susetheseresultstostepthroughadesignand observethenoiseanddistortionthatresultsf romvariousselectionsof(1/ oR2C2).

8 : o=2 1 MHz Q= AV=2(givesanegativegainof2forthesignalpa th)Then, ,wewillusethesinglechannelOPA820 arelativelylow-noise,unity-gainstable, (MHz)en(nV/ Hz)AOL(V/V)in(pA/ Hz) ,givingthisresult: SuggestedR2= PickingR2=250 willallowustoproceedtosettingthe(1/ oR2C2) (usingEquation23)is: MinimumallowedratioofIntegrator/ Thisresultsetsalimittogettingavalidsolut ionforC1 MaximumvaluetogetR3=R2 ThisresultsetsR3=R2fornoisecontrolIftheR 3=R2valueischosen,theresultingzerolocati onsare(fromEquation22): First, Thelowerzerolocationis707kHz + OVERALL NOISE GAINGain (dB) and Phase () 1031041051061071081099060300-30-60-90-12 0-150-180 Frequency (Hz)OPA820 Open-Loop GainOPA820 Open-Loop PhaseNoise-Gain MagnitudeNoise-Gain PhaseExampleDesignsShowingtheImpactofInt egratorPoleLocationContinuingwiththeR3=R 2solution,thefinalcomponentvaluestoplace intoFigure1andthedesignsequenceare.

9 R2=250 [selectedtocontrolnoiseusingEquation9] C2=180pF[setbytargeting(1/ oR2C2)=Q(2AV+1)] C1=1130pF[setusingEquation10] R3=250 [setusingEquationB-17] R1=500 [setusingEquation2] DCGain(AV)= [fromEquation2] FO=1 MHz=F 3dB(ifQ= )[fromEquation3] Q= [fromEquation4]Thenoisegainandphasecanbe computedusingEquation17whereCT= , , (3)= , + OVERALL NOISE GAINGain (dB) and Phase () 1031041051061071081099060300-30-60-90-12 0-150-180 Frequency (Hz)OPA820 Open-Loop GainOPA820 Open-Loop PhaseNoise-Gain MagnitudeNoise-Gain PhaseExampleDesignsShowingtheImpactofInt egratorPoleLocationInordertotesttheimpac tofsettingthe(1/ oR2C2)targetfartoohigh,setitto10andrepea tthedesigninthismanner(stillholdingR2=25 0 ): R2=250 [selectedtocontrolnoiseusingEquation9] C2= [setbytargeting(1/ oR2C2)=10] C1=571pF[setusingEquation10] R3= [setusingEquationB-17] R1= [setusingEquation2] DCGain(AV)= [fromEquation2] FO=1 MHz[fromEquation3] Q= [fromEquation4]Figure6showsthenewdesignc ircuitwiththesemorewidelyseparatednoiseg ainzeroes( ,solvingthenumeratorofEquation19).

10 , , ,thenoisegainpeaksupslightly, AC RESPONSEGain (dB)9630105104106107 Frequency (Hz)Figure 6 Figure ( ) (whileeven-ordertermsdonotcorrelatewellf romsimulationtobenchmeasurements).Also,a 100 (suchasanADCinput) ,VO=2 VPP,RL=100 TESTCIRCUITHD3(dBc)Figure4 ,withthehigherresistorvaluesandpeakednoi segain, (thatisdecreasingfromahighvalueat100Hz) SPOT NOISE VOLTAGEO utput Noise (nV/ )Hz35302520151050103102104105106107 Frequency (Hz)R= (Figure 6)3R= 250 WCircuit(Figure 4) ,theseconddesignpointwiththepeakednoiseg ain(andmuchhigherR3value) ,higherslewrate, , ,sincethebasicMFBcircuitshapesthenoisega intobeaunitygainfeedbackathighfrequencie s, ,usethenon-unity-gainstableOPA2614(adual ) (290 MHzgainbandwidthproduct,orGBW)hasaunity- gainstableversion,theOPA2613(125 MHzGBW), ,thenoisenumbersofthetwoversionsareident ical,butthehighergainbandwidthwillgive20 Log(290/125)= +5 Vsupply,differential-in-to-differential- out, , ( )fora3rd-orderBessellwith5 MHzcutoffgivesthefollowingrequiredpolelo cation.