Download the biquad filter

DIT, Kevin St.
Electric circuits
Waed 3
Chapter 7
The Biquad filter
The biquad configuration is a useful circuit for producing bandpass and low-pass
responses, which require high Q-factor values not achievable with the VSVS and
the IGMF circuits. The Biquad and the state variable filter circuit configuration
can have Q-factor values of 400 or greater. This circuit, whilst not as useful as the
state variable, nevertheless has certain applications. It is easily tuneable using
single resistor tuning (normally a stereo or ganged potentiometer). It can be
configured to produce a Butterworth or a Chebychev response by changing the
damping (1/Q).
Figure 7.47: Biquad filter.
Figure 7.47 shows a typical three-operational amplifier circuit. The transfer
function is obtained by considering the bandpass output first. The low-pass is
easily obtained thereafter. The Biquad active filter consists of a leaky integrator,
an integrator and a summing amplifier. Let Zf be the parallel combination of R2
and C1:
Zf =
R2
1 + sC1 R2
The transfer function for the last stage is defined as:
R
V4
=− 5
V3
R4
7.1
In addition, for the second last stage:
1
V3
=−
V2
sC 2 R3
7.2
Copyright: Paul Tobin School of Electronics and Comms. Eng.
64
DIT, Kevin St.
Electric circuits
Waed 3
The first op-amp is a leaky integrator and we can write for the output voltage as:
Zf
V2 = −
R6
V4 −
Zf
R1
V1
If we substitute for V4 and V2 from equations 7.1 and 7.2, we can write
V4 = −
R5
V3
R4
V3 = −
1
V2
sC 2 R3
But
V4 =
R5 1
V2
R4 sC 2 R3
Substituting 7.6 into 7.3 yields
V2 = −
Zf
Zf
R5
V2 −
V1
R6 sC 2 R3 R4
R1
V2 [1 +
Zf
Zf
R5
]=−
V1
R6 sC 2 R3 R4
R1
Zf
V2
=−
V1
1+
R1
Z f R5
sC2 R3 R4 R6
R2
1 + sC1R2
R1
V2
=−
R2
V1
R5
1 + sC1R2
1+
sC2 R3 R4 R6
R2
R1
V2
=−
V1
1 + sC1 R2 +
R2 R5
sC2 R3 R4 R6
Copyright: Paul Tobin School of Electronics and Comms. Eng.
65
DIT, Kevin St.
Electric circuits
Waed 3
R2
sC2 R3 R4 R6
R1
V2
=− 2
s C1C2 R2 R3 R4 R6 + sC2 R3 R4 R6 + + R2 R5
V1
C 2 R3 R4 R6
R2
s
R1 C1C 2 R2 R3 R4 R6
V2
=−
C 2 R3 R4 R6
R2 R5
V1
s2 + s
++
C1C 2 R2 R3 R4 R6
C1C 2 R2 R3 R4 R6
s
1
C1 R1
V2
=−
R5
1
V1
s2 + s
+
C1 R2 C1C 2 R3 R4 R6
If R4 = R5:
1
C1 R1
s
V2
=−
1
1
V1
s2 + s
+
C1 R2 C1C 2 R3 R6
The standard form for a bandpass second order function is
V2
=−
V1
sK
s2 + s
ωp
Q
ωp
Q
+ωp
2
We may write expressions for the passband edge frequency, Q-factor and gain in
terms of the circuit components by comparing coefficients. The gain at the
resonant frequency is
H (0) = K =
R2
R1
The resonant frequency is
ω 2o =
1
C1C 2 R3 R6
Copyright: Paul Tobin School of Electronics and Comms. Eng.
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DIT, Kevin St.
Electric circuits
Waed 3
C1 R 2 2
Q=
C 2 R6 R3
BW =
1
2πC1 R2
The Q-factor and the resonant frequency are not independent in this circuit. For
high frequencies, the bandwidth will be the same as that for low frequencies. This,
in general, is not a desirable feature. For example, in an audio mixing desk, the
equalising section would use a state-variable circuit where the bandwidth changes
with the higher frequencies. The tuning procedure for the biquad BP is as follows:
1) Select values for C1, C2 and R5,
2) Adjust the resonant frequency ωp by varying R3 set the gain by R1,
3) The Q-factor is set by R2. as opposed to iterative where one has to keep
adjusting the component values to get the desired value, and
4) The Q-factor is set by R2.
Equal value component
If we set C1 = C2 = C and R3 = R6 = R, then the equations are greatly simplified
1
CR1
V2
=−
1
1
V1
s2 + s
+ 2 2
CR2 C R
s
H (0) = K =
R2
R1
ω 2o =
1
C R2
Q=
R2
R
= 2
2
R
R
BW =
1
2πCR2
2
2
Copyright: Paul Tobin School of Electronics and Comms. Eng.
67
DIT, Kevin St.
Electric circuits
Waed 3
Figure 7.48: Biquad filter response.
Copyright: Paul Tobin School of Electronics and Comms. Eng.
68