Handbook for Sound Engineers

Filters and Equalizers 795

(23-47)

where

d is the damping factor,

Rf is the op-amp feedback resistance,

R 0 is the resistance between ground and the inverting
input.

The second-order high-pass filter of Fig. 23-16 is

constructed by reversing the locations of R and C in Fig.

23-15. The gain and damping factor follow the same

equations as for the low pass.

A unity gain Sallen-Key filter can also be made. To

independently control frequency and damping, the ratio

of the capacitors must be changed such that in the low

pass

. (23-48)

The cutoff frequency is still determined by the

product of R and C, so it can be adjusted with the value

of R or by scaling Cf and C 1 together.

Fig. 23-17 is a Sallen-Key filter implemented as a

bipolar junction transistor circuit.

23.3.1.2 State Variable

The state variable filter consists of two low-pass filters

and a summing stage. High-pass, bandpass, and low-

pass outputs are all available from the circuit. The oper-

ation relies on both the magnitude and phase character-

istics of the low-pass sections to generate the outputs.

At high frequency, the low-pass sections attenuate

the signal so that the feedback signal is small, leaving

the unaffected signal at the high-pass output. As the

input frequency approaches the center frequency, the

levels at both the bandpass and low-pass outputs begin

to increase. This leads first to an increase in positive

feedback from the bandpass section giving a damping

dependent overshoot. When the input frequency is

below the center frequency, the net phase shift of both

low-pass sections is 180 degrees, leading to negative

feedback and an attenuation of the high-pass output.

The cutoff frequency of the filter in Fig. 23-18 can

be changed as in the preceding circuits by varying R 1

and R 2 or C 1 and C 2 while keeping other values iden-

tical. The damping factor is varied by changing the

band-pass feedback gain, controlled by the ratio of R 3

and R 4.

(23-49)

The overall gain is controlled by R 12. If R 1 , R2, and

R 12 are equal, the gain is one.

(23-50)

Figure 23-15. Sallen-Key low-pass filter.

Figure 23-16. Sallen-Key high-pass filter.

gain 1

Rf

R 0

----- -+=

12 + – dR 0

R 0

=--------------------------------

= 3 – d

R

Cf C 1

R 0 Rf

• R

R

C C

R 0 Rf

R

Cf^4

d 5

----- -

©¹

§·C

= 1

Figure 23-17. Sallen-Key filter implemented as a Bipolat

Junction Transistor (BJT) circuit.

10 k 7 10 k 7

Cf 16 nF C 1 16 nF

RL

+

d

R 4

R 3

----- -=

gain

R 1

R 12

--------=