DIT, Kevin St. Electric circuits Waed 3

Copyright: Paul Tobin School of Electronics and Comms. Eng. 64

Chapter 7

The Biquad filter

The biquad configuration is a useful circuit for producing bandpass and low-passresponses, which require high Q-factor values not achievable with the VSVS andthe IGMF circuits. The Biquad and the state variable filter circuit configurationcan have Q-factor values of 400 or greater. This circuit, whilst not as useful as thestate variable, nevertheless has certain applications. It is easily tuneable usingsingle resistor tuning (normally a stereo or ganged potentiometer). It can beconfigured to produce a Butterworth or a Chebychev response by changing thedamping (1/Q).

Figure 7.47: Biquad filter.

Figure 7.47 shows a typical three-operational amplifier circuit. The transferfunction is obtained by considering the bandpass output first. The low-pass iseasily 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:

21

2

1 RsC

RZ f +

=

The transfer function for the last stage is defined as:

4

5

3

4

R

R

V

V −= 7.1

In addition, for the second last stage:

322

3 1

RsCV

V −= 7.2

DIT, Kevin St. Electric circuits Waed 3

Copyright: Paul Tobin School of Electronics and Comms. Eng. 65

The first op-amp is a leaky integrator and we can write for the output voltage as:

11

46

2 VR

ZV

R

ZV

ff −−=

If we substitute for V4 and V2 from equations 7.1 and 7.2, we can write

34

54 V

R

RV −=

But

232

3

1V

RsCV −=

2324

54

1V

RsCR

RV =

Substituting 7.6 into 7.3 yields

11

2432

5

62 V

R

ZV

RRsC

R

R

ZV

ff −−=

11432

5

62 ]1[ V

R

Z

RRsC

R

R

ZV

ff −=+

6432

5

1

1

2

1RRRsC

RZR

Z

V

V

f

f

+−=

6432

521

2

1

21

2

1

2

11

1

RRRsC

RRsC

RR

RsC

R

V

V

++

+

−=

6432

5221

1

2

1

2

1RRRsC

RRRsC

R

R

V

V

++−=

DIT, Kevin St. Electric circuits Waed 3

Copyright: Paul Tobin School of Electronics and Comms. Eng. 66

5264326432212

64321

2

1

2

RRRRRsCRRRRCCs

RRRsCR

R

V

V

+++−=

643221

52

643221

64322

643221

6432

1

2

1

2

RRRRCC

RR

RRRRCC

RRRCss

RRRRCC

RRRCs

R

R

V

V

+++−=

64321

5

21

2

11

1

2

1

1

RRRCC

R

RCss

RCs

V

V

++−=

If R4 = R5:

632121

2

11

1

2

11

1

RRCCRCss

RCs

V

V

++−=

The standard form for a bandpass second order function is

221

2

pp

p

Qss

QsK

V

V

ωω

ω

++−=

We may write expressions for the passband edge frequency, Q-factor and gain interms of the circuit components by comparing coefficients. The gain at theresonant frequency is

1

2)0(R

RKH ==

The resonant frequency is

6321

2 1

RRCCo =ω

DIT, Kevin St. Electric circuits Waed 3

Copyright: Paul Tobin School of Electronics and Comms. Eng. 67

362

22

1

RRC

RCQ =

212

1

RCBW

π=

The Q-factor and the resonant frequency are not independent in this circuit. Forhigh 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, theequalising section would use a state-variable circuit where the bandwidth changeswith 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, and4) 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

222

2

1

1

2

11

1

RCCRss

CRs

V

V

++−=

1

2)0(R

RKH ==

22

2 1

RCo =ω

R

R

R

RQ 2

2

22 ==

22

1

CRBW

π=

DIT, Kevin St. Electric circuits Waed 3

Copyright: Paul Tobin School of Electronics and Comms. Eng. 68

Figure 7.48: Biquad filter response.