DESIGN OF a 5th ORDER BUTTERWORTH LPF.docx

DESIGN OF a 5th ORDER BUTTERWORTH LPF.docx

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DESIGNOFa5thORDERBUTTERWORTHLPF

DESIGNOFa5thORDERBUTTERWORTHLOW-PASSFILTERUSINGSALLEN&KEYCIRCUIT

BackgroundTheory:

Filtersareclassifiedaccordingtothefunctionsthattheyaretoperform,intermsofrangesoffrequencies.Wewillbedealingwiththelow-passfilter,whichhasthepropertythatlow-frequencyexcitationsignalcomponentsdowntoandincludingdirectcurrent,aretransmitted,whilehigh-frequencycomponents,uptoandincludinginfiniteonesareblocked.Therangeoflowfrequencies,whicharepassed,iscalledthepassbandorthebandwidthofthefilter.Itextendsfromω=0toω=ωcrad/sec（fcinHz）.Thehighestfrequencytobetransmittedisωc,whichisalsocalledthecutofffrequency.Frequenciesabovecutoffarepreventedfrompassingthroughthefilterandtheyconstitutethefilterstopband.

Theidealresponseofalow-passfilterisshownabove.However,aphysicalcircuitcannotrealizethisresponse.Theactualresponsewillbeingeneralasshownbelow.

Itcanbeseenthatasmallerrorisallowableinthepassband,whilethetransitionfromthepassbandtothestopbandisnotabrupt.

Thesharpnessofthetransitionfromstopbandtopassbandcanbecontrolledtosomedegreeduringthedesignofalow-passfilter.

Theideallow-passfilterresponsecanbeapproximatedbyarationalfunctionapproximationschemesuchastheButterworthresponse.

TheButterworthResponse

NormalizingH0=1and

Then

findingtherootsofD（s）

Example:

Forn=5

Allthepolesare:

-1.0000

-0.8090+0.5878i

-0.8090-0.5878i

-0.3090+0.9511i

-0.3090-0.9511i

0.3090+0.9511i

0.3090-0.9511i

1.0000

0.8090+0.5878i

0.8090-0.5878

POLELOCATIONS

Thepolesaredistributedoverthecircleofradius1（

）.Neverapoleintheimaginaryaxis.

FindingH（s）fromH（s）H（-s）:

H（s）isassignedallRHSpolesH（-s）isassignedallLHSpoles

Followingthisprocedure,theButterworthLPFH（s）（H0=1,wc=1rad/sec）canbefoundforvariousfiltersofordern.

WecanuseMATLABtogetthisdenominatorpolynomial（Butterworthpolynomial）

InMATLAB（code）:

all_poles=roots（[（-1）^n,zeros（1,2*n-1）,1]）

poles=all_poles（find（real（all_poles）<0））

Den=poly（poles）

CircuitDesign:

WewanttodesignofafifthorderButterworthlow-passfilterwithacutofffrequencyof10KHz.

Duringthedesignwemakeuseofmagnitudeandfrequencyscalingandalsooftheuniformchoiceof

asacharacterizingfrequencywillappearinalldesignsteps,exceptforthelastwherethede-normalized（actual）valueswillbefound.

CircuitImplementation:

ImplementationofthecircuitisdoneusingtheSallen&KeyTopology.

thisisofthegeneralform

Ifk=1,

taking

Torealizea5thorderBLPFoneSallen&Keystagewithasingleop-ampisrequiredforeverycomplex-conjugatepolepair.Sincen=5（odd）,anadditionalnegativepoleisrequiredandweuseanRC/voltagefollower.AlsowemadethechoiceofK=1,whichrequiresthattheinvertingop-ampcircuitbereplacedbyavoltagefollowerasshownbelow.

Tofindactualvalues:

Makeallresistors=

Frequencyscaling=

Multiplyingeachcapacitorby

PerformanceMeasures:

Cutofffrequency=10KHz

Frequency（KHz）

Vin（mV）

Vout（mV）

1.502

500

498.5

2.009

500

493.5

5.782

500

493.7

9.001

500

487

9.6

500

387.5

10.04

500

245.7

11.01

500

225.7

12.04

500

187.5

13.02

500

115.6

14.02

500

90.62

15.06

500

75.00

16.01

500

62.50

17.01

500

47.50

Idealresponse:

Actualresponse:

Fromtherecordedvaluesaftermeasurements.

MeasureddBgainvaluesvs.logfrequencyvalues

CircuitDiagram:

FinalCircuit

Parts:

Part

Quantity

LM324N

LowPowerQuadOperationalAmplifier

1

1KΩresistor

5

0.016μFcapacitor

1

0.019μFcapacitor

1

0.013μFcapacitor

1

0.051μFcapacitor

1

0.0049μFcapacitor

1

9Vbattery

2

References:

DeliyannisT.,YichuangSun,andJ.K.Fidler1999,Continuous-timeActiveFilterDesign,CRCPress,NewYork.

VanValkenburgM.E.,AnalogFilterDesign,1982,OxfordUniversityPress,NewYork.

ChenWai-Kai,Passive&ActiveFilterDesign,Chapter2,pp.50-92

HuelsmanL.P.andP.E.Allen,IntroductiontotheTheoryandDesignofActiveFilters