Design an analog Butterworth lowpass filter with the following specifications:

Question

Passband cutoff frequency = fp = 1225 Hz Stopband cutoff frequency = fs = 2725 Hz Maximum attenuation in the passband (dB) = Ap = 1.07 dB Minimum attenuation in the stopband (dB) = As = 12.32 dB excess='stop' km = 1000 kf = ?c?p

(a) Find ?p, ?s, ?s, and order n
(b) Find the recalculated values of Ap1, As1, ?, ?c.
(c) Find the poles S and zeros SZ of the normalized lowpass filter (do not multiply by ?c)
(d) Find the transfer function of the normalized lowpass filter and find Q values of the second- order sections.
(e) Find the scaled (both magnitude and frequency) component values when unity gain Sallen and Key circuit is used for each second order section to implement the filter.

Answer 1

(a) ?p = 7696.902 rad/s, ?s = 17121.68 rad/s, ?s = 2.2245, n = ceil (2.53) = 3
(b) Ap1 = 1.07 dB, As1 = 15.4222 dB, ? = 0.5286, ?c = 1.2368 rad/s
(c) Poles: S = -1, S = -0.5 + j0.866, S = -0.5 - j0.866 Zeros: SZ = ?, SZ = ?, SZ = ?
(d) 2 1 1 ( ) 1 1 H S S S S = × + + + Q = 1 for the second-order term.
(e) kf = ?o = ?c?p = 9519.5496 rad/s Second-order section: Normalized: C1 = 2Q = 2 F, C2 = 1/(2Q) = 0.5 F, R1 = 1 ?, R2 = 1 ? Scaled: C1 = 0.21 ?F, C2 = 52.5235 nF, R1 = 1 k?, R2 = 1 k? First-order section: Normalized: R = 1 ?, C = 1 F Scaled: R = 1 k?, C = 0.105 ?F