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OPAMP REALIZATIONS OF BUTTERWORTH FILTERS
- In designing passive high-order filters the interaction of cascaded filter stages must be taken
into account since each stage presents a load to preceding and following stages that can vary
the design parameters of the filter.
- Individual OpAmp circuits can be placed in cascade without interaction of the individual
stages: each stage does not typically present a gain-changing load to either preceding or
following stages.
A
VT
(^ ω ) =^ A
Vi
(^ ω )
i = 1
N
∏.^ (9.4-1)
- The OpAmp circuits must meet the following design criteria:
◊ The resonant frequency, ω o
, must be variable in both first and second order stages.
◊ The damping coefficient, ζ, of second order stages must be variable.
- For Butterworth low-pass and high-pass filters, ω o
is the 3 dB frequency and ζ is the
damping coefficient (tabulated in Table 9.3-1).
Low-pass OpAmp Filters:
- First and second order low-pass voltage transfer relationships.
R
R R'
C
1
1
v
v i
o
−
(a)
First Order Low-pass
R R'
R
C
1
1
v
v i
R o
C
2
2
−
v b
(b)
Second Order Low-pass
Figure 9.4-1 Low-pass Section Realizations
First order low-pass stage
- The transfer characteristic is:
is determined by the input RC time constant:
- Low-frequency gain of the circuit is adjustable through the elements R ' and R :
Second order low-pass stage
- The second order stage (Figure 9.4-1b) was developed by R. P. Sallen and E. L. Key and is
therefore known as the Sallen and Key circuit.
- The transfer characteristic is:
v o
v i
R ' R
1 + j ω R 1
+ R
C
1
R '
( R ) R 2 C 2
+ ( j ω)
2 R 1
R
2
C
1
C
- Low-frequency gain of the circuit is adjustable through the elements R ' and R :
A
Vo
R ' R ,^ (9.4-8)
is determined by:
and
2 ζ
ω 0
= R
1
+ R
C
1
− R '^
( R )
R
2
C
2
- Three constraints on the design and five quantities to vary.
- For R 1
= R
2
= R
C
and C 1
= C
2
= C
C
, we have the uniform time constant design :
and ω o
R
C
C
C
Unity gain designs:
- For filters with unity gain in the passband:
A
Vo
R' R
- The transfer relationship is:
v o
v i
1 + j ω R 1
+ R
C
1
+ ( j ω)
2 R 1
R
2
C
1
C
- The pertinent design parameters are:
and
2 ζ
ω 0
= R
1
+ R
C
1
Design Procedure for OpAmp Butterworth Filters (Uniform Time Constant)
- Determine the class of filter (Butterworth, Chebyshev, Elliptic, Bessel, etc.)
- Determine the type of filter (low-pass, high-pass, bandpass, bandstop)
- Determine the order of the filter
- Determine the appropriate coefficients: for Butterworth, find ω o
and damping
coefficients, ζ as appropriate.
- Choose standard capacitor values for C 1
and C 2
- Find R 1
, R
2
, R , and R'.
High-pass OpAmp Filters:
- The two low-pass circuits of Figure 9.4-1 can be converted into high-pass filter sections
simply by interchanging the position of the numbered capacitors with the numbered resistors.
Interchanging these elements retains the same number of transfer function poles and adds
zero-frequency zeroes.
Unity gain designs:
- For filters with unity gain in the passband:
A
Vo
R ' R =^0 (9.4-25)
- The transfer relationship is:
v o
v i
( j^ ω)
2 R 1
R
2
C
1
C
1 + j ω R 2
C
1
+ C
+ ( j ω)
2 R 1
R
2
C
1
C
- The pertinent design parameters are:
ω 0
R
1
R
2
C
1
C
2
and
2 ζ
ω 0
= R
2
C
1
+ C
Band-pass and Band-stop OpAmp Filters
- In many cases band-pass and band-stop filters can be achieved with the series or parallel
connection of high-pass and low-pass filters.
- The series connection of a low-pass filter with a high-pass filter will produce a band-pass
filter if there exists a region of common passband.
Passband
0
0
2
ω
|H( )|ω
ω 1
ω
Overall
Low Passband
High Passband
Figure 9.4-3 Cascaded Band-pass Filter Characteristic
- The parallel connection of a low-pass filter and a high-pass filter will produce a band-stop
filter if there is a region of common stopband.
Stopband
0
0
2
ω
|H( )|ω
ω 1
ω
Overall
Low High
Passband Passband
Figure 9.4-4 Parallel Band-stop Filter Characteristic
