DESIGN OF a 5th ORDER BUTTERWORTH LPF文档格式.docx
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DESIGN OF a 5th ORDER BUTTERWORTH LPF文档格式.docx
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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
493.5
5.782
493.7
9.001
487
9.6
387.5
10.04
245.7
11.01
225.7
12.04
187.5
13.02
115.6
14.02
90.62
15.06
75.00
16.01
62.50
17.01
47.50
Idealresponse:
Actualresponse:
Fromtherecordedvaluesaftermeasurements.
MeasureddBgainvaluesvs.logfrequencyvalues
CircuitDiagram:
FinalCircuit
Parts:
Part
Quantity
LM324N
LowPowerQuadOperationalAmplifier
1
1KΩresistor
5
0.016μFcapacitor
0.019μFcapacitor
0.013μFcapacitor
0.051μFcapacitor
0.0049μFcapacitor
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
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