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Transcript of Active Filters
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)
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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.
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
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
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 !