Second Order Filters

Flashcards about second order filters.

General second order filter response

Generic filter response and its characteristics. H(jω) = N(jω) / (1 − (ω/ω0)2 + (jω/ω0)/Q)

Low-pass filter response

HLP F (jω) = 1 / (1 − (ω/ω0)2 + (jω/ω0)/Q), |HLP |max = Q * sqrt(1 − 1/4Q2)

High-pass filter response

HHP F (jω) = −(ω/ω0)2 / (1 − (ω/ω0)2 + (jω/ω0)/Q

Band-pass filter response

HBP F (jω) = (jω/ω0)/Q / (1 − (ω/ω0)2 + (jω/ω0)/Q, ωL = ω0 * sqrt(1 + 1/4Q2 − 1/2Q), ωH = ω0 * sqrt(1 + 1/4Q2 + 1/2Q), ω0 = sqrt(ωL*ωH), Q = ω0 / BW

Sallen-Key (KRC) configuration (Low-pass filter)

A second order filter configuration. Transfer function: H(jω) = K / (1 − ω2R1C1R2C2 + jω [(1 − K) R1C1 + R1C2 + R2C2])

Sallen-Key (KRC) configuration (Low-pass filter) Standard design equations

H0LP = K = 1 + RB/RA, ω0 = sqrt(1 / R1C1R2C2), Q = 1 / ((1−K)*sqrt(R1C1/R2C2)+sqrt(R1C2/R2C1)+sqrt(R2C2/R1C1))

Sallen-Key (KRC) configuration (Low-pass filter) Simplification: Unity gain

H0LP = 1 V/V, R2 = R, R1 = mR, C2 = C, C1 = nC, ω0 = sqrt( 1 / mnRC), Q = sqrt(mn) / (m+1), n ≥ 4Q2, m = k + sqrt(k2 − 1), k = (n/2Q2) − 1, Modification: Independent (of Q) gain Anew = (R1B / R1A+R1B) * Aold, R1 = R1A||R1B, R1A = R1 * (Aold / Anew), R1B = R1 / (1−Anew/Aold)

Sallen-Key (KRC) configuration (High-pass filter) Design equations

H0HP = K, ω0 = sqrt(1 / R1C1R2C2), Q = 1 / ((1−K)*sqrt(R2C2/R1C1)+sqrt(R1C2/R2C1)+sqrt(R1C1/R2C2))

Sallen-Key (KRC) configuration (Band-pass filter) Design equations

H0BP = K / (1+(1−K)R1/R3+(1+C1/C2)R1/R2), ω0 = sqrt(1+R1/R3) / sqrt(R1C1R2C2), Q = sqrt(1+R1/R3) / ([1+(1−K)R1/R3]sqrt(R2C2/R1C1)+sqrt(R1C2/R2C1)+sqrt(R1C1/R2C2))

Sallen-Key (KRC) configuration (Band-pass filter) Simplified design equations

Q > sqrt(2)*3, R1 = R2 = R3 = R, C1 = C2 = C, H0BP = K/(4−K), ω0 = sqrt(2) / RC, Q = sqrt(2) / (4−K)

Sallen-Key (KRC) configuration (Notch filter) Design equations

H0NF = K, ω0 = 1 / RC, Q = 1 / (4−2K)

Multiple feedback configuration (Band-pass filter) Design equations

H0BP = −R2/R1 / (1+C1/C2), ω0 = sqrt(1 / R1C1R2C2), Q = sqrt(R2/R1) / (sqrt(C1/C2)+sqrt(C2/C1)), Design for |H0BP | = H0 < 2Q2: R1A = Q/H0ω0C, R1B = R1A / ((2Q2/H0) − 1)

Multiple feedback configuration (Low-pass filter) Design equations

H0LP = − R3 / R1, ω0 = sqrt(1 / R2C1R3C2), Q = sqrt(C1/C2) / sqrt(R2R3/R21)+sqrt(R3/R2)+sqrt(R2/R3), One approach: C1 = nC2, n ≥ 4Q2 * (1 + |H0LP |), R3 = (1+sqrt(1−4Q2(1+|H0LP |)/n)) / (2ω0QC2), R1 = R3 / |H0LP |, R2 = 1 / (ω20R3C1C2)

Multiple feedback configuration (Notch filter) Design equations

Design a BPF with negative output (like the MFB BPF), Design a summation amplifier

State variable filter Design equations

ω0 = sqrt(R5/R4) / sqrt(R6C1R7C2), Q = (1+R2/R1)*sqrt(R5R6C1/R4R7C2) / (1+R5/R3+R5/R4), H0HP = − R5 / R3, H0LP = − R4 / R3, H0BP = (1+R2/R1) / (1+R3/R4+R3/R5)

State variable filter topology Design equations (simplified with mostly equal component)

R5 = R4 = R3, R6 = R7 = R, C1 = C2 = C, ω0 = 1 / RC, Q = 1 / (3 * (1 + R2/R1)), H0HP = −1, H0LP = −1, H0BP = Q

Biquad filter topology Design equations

ω0 = sqrt(1 / R4C1R5C2), Q = R2 / sqrt(R4R5C1C2), H0LP = R5 / R1, H0BP = − R2 / R1

Biquad filter topology Simplified

R5 = R4 = R, C1 = C2 = C, ω0 = 1 / RC, Q = R2/R, H0LP = R / R1, H0BP = − R2 / R1

Biquad filter topology (Notch design)

Similar approach to MFB notch filter design; Use a BPF; Use a summation amplifier; Synthesise vi − vBP, Simplified (symmetric notch): ωz = ω0 = 1 / RC, Q = R1 / R, H0N = − R5 / R2