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EEE C364/INSTR C364Analog Electronics
First order Filter
Lecture 2117-03-2012
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Second order Passive and Active low pass filter
At low frequencies H 1V/V At high frequencies a single RC
section gives H 1/j( / 0) At high frequencies a two stage RC
section gives H 1/j( / 0) 1/j( / 0)=1/j( / 0)2,
21
01
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In fig (b) there will be considerable positive feedback only in
the vicinityof = 0 .For >> 0 , the shunting action of C2
makes output V0 too smallFilters of this type are called KRC
filters or Sallen-Key filters, for theirinventors.
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Sallen-Key (VCVS) Filters
)()1()(
Vthatso KVVBut
)(
VnodeatKCL
)()()(
V NodeatKCL
32143221
210
02
41120
42221
2
30122111
1
Y Y Y Y Y Y K Y Y Y KY
V V
s H
V KY
Y Y
Y V Y V V
Y V V Y V V Y V V
S
S
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VCVS Low-pass filter
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VCVS Low-pass filter
1.4142k filter,h butterwortaforfunctiontransferstandardwith
thecomparing
CCR R CR K -1
CR 1
CR 1
2k and CCR R
1 where
2)(
sCY ,sCY ,R 1
Y ,R 1
Y ,1
)()1()(
2/12121
2212112121
2
22
20
24132
21
1
32143221
210
c
cc
c
S
A
B
S
sk s K
V V
s H
R R
K
Y Y Y Y Y Y K Y Y Y KY
V V
s H
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Impedance Scaling: To scale impedance while maintaining a
constantcorner frequency, use the following factor
Frequency scaling: To scale frequency while maintaining a
constantimpedance, use the following factor
C
K old
new
K K 1
Cnewand R K R
CandR of valuesnewthefactors,twoCombining
changenohavewillR ,C/K tochangewillCSo
ZZnew
ZK C
toCandKZR tochangewillR So
old
new Z Z
Z K
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High pass filter
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High pass filter
1.4142k filter,h butterwortaforfunctiontransferstandardwith
thecomparing
CCR R CR K -1
CR 1
CR 1
2k and CCR R
1 where
2)(
)()1()(
2/12121
2212112121
2
22
20
32143221
210
c
ccS
S
sk s Ks
V V
s H
Y Y Y Y Y Y K Y Y Y KY
V V
s H
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VCVS Band-Pass filter
----------------------------------------------------------------------------To
transform normalized Substitute for s in T(s)low-pas filter
to-----------------------------------------------------------------
L H
L H
L H
L H
c
s s s
s s
2
2
)( filterreject-Band
)(
filterPass-Band
filterPassHigh
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VCVS Band-Pass filter
1sQ1s
sQT
T(s)or
1BWssBWsTT(s)getwe
1sT
T(s)functionnsferfilter tra passlowin thesReplacing
1for1f
QfactorqualityThe
1
BW
1
aswritten becanationtransformthe,1For
BWanddefineWe
)(
1,hfilter wit passlowof expressionin the s substituteWe
2
0
20
0
00
0
20
2
c
BW BW
s s
s s
L H L H
L H
L H
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VCVS Band-Pass filter
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VCVS Band-Pass filter
)/R R (1K :gainAmplifier
K)-1/(3Q:factor-Q
K)-K/(3T:f atGain
RC)1/(2f frequencycentreThe
1sQ
1s
sQT
T(s)equationwith thecomparing 1)3(
)(
AB
00
0
2
0
022
022
0
s K RC sC R s KRC
sT
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VCVS Band-reject filter
1sQ1s
1)(sTT(s)or
1BWss)1(T
T(s)getwe
1sT
T(s)functionnsferfilter tra passlowin thesReplacing
1for1f
QfactorqualityThe
1B
aswritten becanationtransformthe,1For
BWanddefineWe
)(
1,hfilter wit passlowof expressionin thessubstituteWe
2
20
2
20
0
00
0
20
2
c
s
BW BW
s s
W
s
s
L H L H
L H
L H
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VCVS Band-reject filter
)/R R (1K :gainAmplifier
K)-2(21
Q:factor-Q
KT:f atGainRC)1/(2f frequencycentreThe
1sQ
1s
1)(sTT(s)equationwith thecomparing
1)2(2)1(
)(
AB
00
0
2
20
2
2
s K s s K
sT
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VCVS all-pass filter
i
ii
i iii
ii
c
c
i i
j j
iii
iii
a
a s frequencie
bbaba
T
b
ee sb sa
sb sa
sb sa sb sa
ator deno sb sa
i
gr
4222
2
gr c
gr gr gr
gr
2i1-
22
2
211
211
0211
0
bQasdefinedisfactorqualityThe
1T0),(lowAt
)2(1)1(1
T 1,For
dd212tt
Tdelay,group Normalized
dd
-t by,givenisdelaygroup
1a
tan-2-2where
1)1(
)1(T(s)
aswrittenisfilter passallanof functiontransfergeneralThe
11
T(s)
getwe,mincomplexconjugatewith theTthereplacing 1
TT(s)
filter pass-lowaof functiontransfergeneralheConsider t
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