ACTIVE FILTERS...FROM MIDDLEHURST: PRACTICAL FILTER DESIGN.
THERE IS A TOPOLOGY FOR LP AND HP ACTIVE FILTERS THAT USES THREE R-C S WITH ONE OP-AMP TO GET THREE POLES IN THE TRANSFER FUNCTION. THIS TOPOLOGY IS USED FOR SALLEN-KEY FILTERS AND IS CALCULATED FOR BUTTERWORTH FILTERS IN FIG. 2.7 IN THE BOOK.

LOW PASS TOPOLOGY

                     ----C1---------------
                     I                   I
                     I                   I
V(IN)----R--------R--------R--------A=1-------- V(OUT)
             I               I
             I               I
             C3              C2
             I               I
             I               I
             -               -


USING EQUAL RESISTORS R, WE CAN FIND THE THREE CAPACITORS. THIS IS DONE USING THE PROGRAMS IN 2.4 AND 2.7 FOR THE BUTTERWORTH CASE. IN GENERAL, WE SHOULD ALSO BE ABLE TO DO IT, FOR CHEBYCHEV FILTERS FOR EXAMPLE.

HERE THE TRANSFER FUNCTION G(S)=V(OUT)/V(IN) IS

G(S) = 1 / (1 + (S)*(3*C2+C3)*R + (S^2)*(C1*C2+C2*C3)*R^2 +
(S^3)*(C1*C2*C3)*R^3)

OR G(S) = 1 / (1 + (S)*A1 + (S^2)*A2 + (S^3)*A3)

or G(JW) = 1 / (1 + (JW)*A1 - (W^2)*A2 - (JW)*(W^2)*A3)

OR |G(JW)|^2 = 1 / [(1 - (W^2)*A2 + J*(W*A1 - (W^3)*A3)) *
(1 -(W^2)*A2 - J*(W*A1 - (W^3)*A3))]

OR |G(JW)|^2 = 1 / [[1 - (W^2)*A2]^2 + [(W*A1 - (W^3)*A3]^2]

SO FOR NORMALIZED BUTTERWORTH FILTERS (WC = 1)
WHERE G MUST BE |G(JW)|^2 = 1 / (1 + W^2*3)

MAKES A3 = 1 2*A2 = A1^2 A2^2 = 2*A1*A3

OR A1 = 2 A2 = 2 A3 = 1 FOR BW LP FILTERS

EXAMPLE: 3-P BW LPF WITH FC = 3400 HZ (WC = 21362).
LET R = 10K. THEN C1 = 16.6 NF, C2 = 0.95 NF, C3 = 6.5 NF.
NOTE THAT IT IS ABOUT 40 DB DOWN AT 16 KHZ.
J. E. MATZ - 10 MAR 97