Active Filters
-
Upload
allen-kerr -
Category
Documents
-
view
54 -
download
4
description
Transcript of Active Filters
![Page 1: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/1.jpg)
Ch. 12 Active Filters Part 1 1ECES 352 Winter 2007
Active Filters
* Based on use of amplifiers to achieve filter function
* Frequently use op amps so filter may have some gain as well.
* Alternative to LRC-based filters* Benefits
Provide improved characteristics
Smaller size and weight Monolithic integration in IC Implement without inductors Lower cost More reliable Less power dissipation
* Price Added complexity More design effort
dBin)T(20log)A(
1)A(for )A(function n Attenuatio
or
dBin)T(20log)G(
1)A(for )G(function Gain
:as expressed becan Magnitude
)(
)(
sV
sVsT
i
o
Transfer Function
Vo(s)Vi(s)
![Page 2: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/2.jpg)
Ch. 12 Active Filters Part 1 2ECES 352 Winter 2007
Filter Types* Four major filter types:
Low pass (blocks high frequencies)
High pass (blocks low frequencies)
Bandpass (blocks high and low frequencies except in narrow band)
Bandstop (blocks frequencies in a narrow band)
Low Pass High Pass
Bandpass Bandstop
![Page 3: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/3.jpg)
Ch. 12 Active Filters Part 1 3ECES 352 Winter 2007
Filter Specifications
* Specifications - four parameters needed Example – low pass filter: Amin, Amax, Passband, Stopband
Parameters specify the basic characteristics of filter, e.g. low pass filtering Specify limitations to its ability to filter, e.g. nonuniform transmission in
passband, incomplete blocking of frequencies in stopband
![Page 4: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/4.jpg)
Ch. 12 Active Filters Part 1 4ECES 352 Winter 2007
Filter Transfer Function
* Any filter transfer function T(s) can be written as a ratio of two polynomials in “s”
* Where M < N and N is called the “order” of the filter function Higher N means better filter performance Higher N also means more complex circuit implementation
* Filter transfer function T(s) can be rewritten as
where z’s are “zeros” and p’s are “poles” of T(s) where poles and zeroes can be real or complex
* Form of transfer function is similar to low frequency function FL(s) seen previously for amplifiers where A(s) = AMFL(s)FH(s)
oN
NN
oM
MM
M
bsbs
asasasT
....
....)(
11
11
N
MM
pspsps
zszszsasT
....
....)(
21
21
![Page 5: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/5.jpg)
Ch. 12 Active Filters Part 1 5ECES 352 Winter 2007
First Order Filter Functions
0
11
0
1)(
s
a
asa
s
asasT
o
o
* First order filter functions are of the general form
Low Pass
High Pass
a1 = 0
a0 = 0
![Page 6: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/6.jpg)
Ch. 12 Active Filters Part 1 6ECES 352 Winter 2007
First Order Filter Functions* First order filter functions are of the form
0
11
0
1)(
s
a
asa
s
asasT
o
o
General
All Pass
a1 0,a2 0
![Page 7: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/7.jpg)
Ch. 12 Active Filters Part 1 7ECES 352 Winter 2007
Example of First Order Filter - Passive* Low Pass Filter
00
0
2
2/1
2
001
0
0
0
log20,For
01,For
1log10
1
1log20)(T
0a
thfilter wi pass) (loworder first a of form thehas This
1
1
1
)/1(
11
1
1
11
1
)(
)()(
T
dBT
dBin
a
RCwhere
ss
RCs
sRCsC
R
sCRZ
Z
sV
sVsT
o
o
o
C
C
i
o
0 dB
![Page 8: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/8.jpg)
Ch. 12 Active Filters Part 1 8ECES 352 Winter 2007
20 log (R2/R1)
Example of First Order Filter - Active* Low Pass Filter
01
20
121
20
2
1
2
2/1
21
2
01
201
2
0
0
1
2
0
1
2
2
1
2
21
2
1
2
1
2
11
2
log20log20,For
0log20,For
1log10log20
1
1log20log20)(T
0a
thfilter wi pass) (loworder first a of form thehas This
1
1
1
)/1(
11
1
)1(
)/1(
1
)(
)()(
R
RT
RRfordBR
RT
R
R
R
RdBin
R
Ra
CRwhere
sR
RsR
R
CRsR
R
CsRR
R
R
sCR
R
ZR
RI
ZRI
sV
sVsT
o
o
o
CCo
i
o
V_= 0
Io
I1 = Io
Gain Filter function
![Page 9: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/9.jpg)
Ch. 12 Active Filters Part 1 9ECES 352 Winter 2007
Second-Order Filter Functions
o
o
bsbs
asasasT
1
21
22)(
* Second order filter functions are of the form
which we can rewrite as
where o and Q determine the poles
* There are seven second order filter types:Low pass, high pass, bandpass, notch,Low-pass notch, High-pass notch andAll-pass
20
02
12
2)(
sQ
s
asasasT o
2
0
02121 41
22,, Q
Qj
Qpp PP
js-plane
o
x
x
Qo
2
This looks like the expression for the new poles that we had for a feedback amplifier with two poles.
![Page 10: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/10.jpg)
Ch. 12 Active Filters Part 1 10ECES 352 Winter 2007
Second-Order Filter Functions
Low Pass
High Pass
Bandpass
a1= 0, a2= 0
a0= 0, a1= 0
a0= 0, a2= 0
20
02
12
2)(
sQ
s
asasasT o
![Page 11: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/11.jpg)
Ch. 12 Active Filters Part 1 11ECES 352 Winter 2007
Second-Order Filter Functions
Notch
Low Pass Notch
High Pass Notch
a1= 0, ao = ωo2
a1= 0, ao > ωo2
a1= 0, ao < ωo2
20
02
12
2)(
sQ
s
asasasT o
All-Pass
![Page 12: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/12.jpg)
Ch. 12 Active Filters Part 1 12ECES 352 Winter 2007
Passive Second Order Filter Functions
* Second order filter functions can be implemented with simple RLC circuits
* General form is that of a voltage divider with a transfer function given by
* Seven types of second order filters High pass Low pass Bandpass Notch at ωo
General notch Low pass notch High pass notch
20
02
12
2
)(
)()(
sQ
s
asasa
sV
sVsT o
i
o
![Page 13: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/13.jpg)
Ch. 12 Active Filters Part 1 13ECES 352 Winter 2007
* Low pass filter
Example - Passive Second Order Filter Function
20
02
12
2
)(
)()(
sQ
s
asasa
sV
sVsT o
i
o
General form of transfer function
20012
22
2
22
0a,0a
thfilter wiorder second a of form thehas This
1
11
1
1
1
11
1
1
1)(
111
111
)(
)()(
a
L
CRRCQand
LCwhere
sQ
sLCRC
ss
LC
LCsRLs
sRCR
sLRsRC
sRCR
sL
sRCR
ZRZ
RZsT
sosRC
R
RsC
RZ
RZwhere
ZRZ
RZ
sV
sVsT
oo
oo
o
LC
C
C
C
LC
C
i
o
T(dB)
0
01
0
)(22
2
sas
jsasQ
sas
sQ
ssT o
oo
o
0 dBQ
![Page 14: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/14.jpg)
Ch. 12 Active Filters Part 1 14ECES 352 Winter 2007
Example - Passive Second Order Filter Function* Bandpass filter
01
a,0a
thfilter wiorder second a of form thehas This
1
1
11
1
11
1)(
111
111
)(
)()(
012
222
2
2
2
2
aQRC
L
CRRCQand
LCwhere
sQ
s
sRC
LCRCss
RCs
LCsRsL
sL
LCs
sLR
LCs
sL
RZZ
ZZsT
soLCs
sL
sLsC
ZZ
ZZwhere
RZZ
ZZ
sV
sVsT
o
oo
oo
LC
LC
LC
LC
LC
LC
i
o
20
02
12
2
)(
)()(
sQ
s
asasa
sV
sVsT o
i
o
General form of transfer function
00
1
0
)(
2
2
22
0
sass
jsat
sass
s
sQ
s
Qs
sT
o
o
oo
T(dB)
0
0 dB
-3 dB
Qo
![Page 15: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/15.jpg)
Ch. 12 Active Filters Part 1 15ECES 352 Winter 2007
Single-Amplifier Biquadratic Active Filters* Generate a filter with second order
characteristics using amplifiers, R’s and C’s, but no inductors.
* Use op amps since readily available and inexpensive
* Use feedback amplifier configuration Will allow us to achieve filter-like
characteristics
* Design feedback network of resistors and capacitors to get the desired frequency form for the filter, i.e. type of filter, e.g bandpass.
* Determine sizes of R’s and C’s to get desired frequency characteristics (0 and Q), e.g. center frequency and bandwidth.
* Note: The frequency characteristics for the active filter will be independent of the op amp’s frequency characteristics.
Example - Bandpass Filter
20
02
12
2
)(
)()(
sQ
s
asasa
sV
sVsT o
i
o
General form of transfer function
![Page 16: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/16.jpg)
Ch. 12 Active Filters Part 1 16ECES 352 Winter 2007
Design of the Feedback Network
* General form of the transfer function for feedback network is
* Loop gain for feedback amplifier is
* Gain with feedback for feedback amplifier is
* Poles of feedback amplifier (filter) are found from setting
)(
)()(
sD
sN
V
Vst
b
a
)(
)()(
sD
sNAAstAf
Ast
A
A
AA
ff )(11
0)(
)()(
sincefilter theof poles theion)approximat good a (to
becomecircuit feedback from t(s)of zeros theSo
1since01
)(
0)(1
sD
sNst
AA
st
orAst
Conclusion: Poles of the filter are the same as the zeros of the RC feedback network !
Design Approach: 1. Analyze RC feedback network to find expressions for zeros in terms R’s and C’s.2. From desired 0 and Q for the filter, calculate R’s and C’s. 3. Determine where to inject input signal to get desired form of filter, e.g. bandpass.
![Page 17: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/17.jpg)
Ch. 12 Active Filters Part 1 17ECES 352 Winter 2007
Design of the Feedback Network* Bridged-T networks (2 R’s and 2C’s)
can be used as feedback networks to implement several of the second order filter functions.
* Need to analyze bridged-T network to get transfer function t(s) of the feedback network. We will show that
* Zeros of this t(s) will give the pole frequencies for the active filter..
4321413231
2
4321321
2
1111
1111
)(
RRCCRCRCRCss
RRCCRCCss
V
Vst
b
a
Bridged – T network
b
a
V
Vst )(
20
02
12
2
)(
)()(
sQ
s
asasa
sV
sVsT o
i
o
General form of filter’s transfer function
![Page 18: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/18.jpg)
Ch. 12 Active Filters Part 1 18ECES 352 Winter 2007
Analysis of t(s) for Bridged-T Network
Vb
Va
I3 = (Vb-Va)/R3
I2 = I3
Ia = 0I1
for t(s)result final get the wegRearrangin
11
11111
1
11111
1
11
11
11
1
. and of in terms and find Now
V and V of in terms I and I findingby Begin
)(
2323
2433132341
1211
24333234
3243234241
2432344
124
2323232212
41
332
33
ba32
CsR
V
CsRV
CRsRRsC
V
RCsRRsC
V
VZIV
CRsRRV
RCsRRV
R
VV
CRsR
V
CsRR
VIII
CRsR
V
CsRR
V
R
VI
CsR
V
CsRV
sCR
VVVZIVV
VVII
R
VVIIand
R
VVI
V
Vst
ba
ba
Cb
ba
abba
ba
ba
abaCa
ba
abab
b
a
V12
I4
Analysis for t(s) = Va / Vb
4321413231
2
4321321
2
1111
1111
)(
RRCCRCRCRCss
RRCCRCCss
V
Vst
b
a
![Page 19: Active Filters](https://reader036.fdocuments.us/reader036/viewer/2022081421/56812a9b550346895d8e538c/html5/thumbnails/19.jpg)
Ch. 12 Active Filters Part 1 19ECES 352 Winter 2007
4321413231
2
4321321
2
1111
1111
)(
RRCCRCRCRCss
RRCCRCCss
V
Vst
b
a
Analysis of Bridged-T Network* Setting numerator of t(s) = 0 gives zeroes
of t(s), which are also the poles of filter’s transfer function T(s) since
* Where the general form of filter’s T(s) is
* Then comparing the numerator of t(s) and the denominator of T(s), o and Q are related to the R’s and C’s by
* so
* Given the desired filter characteristics specified by o and Q, the R’s and C’s can now be calculated to build the filter.
20
02
12
2
)(
)()(
sQ
s
asasa
sV
sVsT o
i
o
4321
1
RRCCo
321
111
RCCQo
4
321
2121
213
4321
11
R
RCC
CCCC
CCR
RRCCQQ
o
o
.0)(
1)()(
)(11)(
)(
)(
sAstwhensTASo
Ast
A
A
AsT
sV
sVA
f
fi
of
These have the same form – a quadratic !