Second Order Filters

General Second Filter Responses

• Topics:
  • General second filter responses
  • Second order filters configurations
  • Sallen-Key filters (KRC)
  • Multiple feedback filters
  • State variable filters
  • Biquad filters

Common Second Order Filter Responses

• Generic filter response and its characteristics
  • General second order filter response: H(jω) = \frac{N(jω)}{1 - (ω/ω0)^2 + (jω/ω0)/Q}
  • Low-pass filter response: H_{LPF}(jω) = \frac{1}{1 - (ω/ω0)^2 + (jω/ω0)/Q}
    • |H{LP}|{max} = Q \sqrt{1 - 1/4Q^2}
  • High-pass filter response: H_{HPF}(jω) = \frac{-(ω/ω0)^2}{1 - (ω/ω0)^2 + (jω/ω0)/Q}
  • Band-pass filter response: H_{BPF}(jω) = \frac{(jω/ω0)/Q}{1 - (ω/ω0)^2 + (jω/ω0)/Q}
    • \omegaL = \omega0 \sqrt{1 + \frac{1}{4Q^2}} - \frac{1}{2Q}, \omegaH = \omega0 \sqrt{1 + \frac{1}{4Q^2}} + \frac{1}{2Q}
    • \omega0 = \sqrt{\omegaL\omegaH}, Q = \frac{\omega0}{BW}

Second Order Filter Configurations

• Sallen-Key (KRC) Configuration (Low-pass filter)
  • Transfer function: H(jω) = K \frac{1}{1 - ω^2R1C1R2C2 + jω [(1 - K) R1C1 + R1C2 + R2C2]}
  • Standard design equations:
    • H{0LP} = K = 1 + \frac{RB}{R_A}
    • \omega0 = \sqrt{\frac{1}{R1C1R2C_2}}
    • Q = \frac{1}{(1-K)\sqrt{R1C1/R2C2}+\sqrt{R1C2/R2C1}+\sqrt{R2C2/R1C1}}
  • Simplification: Equal components (R = R1 = R2, C = C1 = C2)
    • H{0LP} = K = 1 + \frac{RB}{R_A}
    • \omega_0 = \frac{1}{RC}
    • Q = \frac{1}{(3-K)}
  • Simplification: Unity gain (H_{0LP} = 1 V/V)
    • R2 = R, R1 = mR, C2 = C, C1 = nC
    • \omega_0 = \sqrt{\frac{1}{mnR^2C^2}} = \frac{1}{RC\sqrt{mn}}
    • Q = \frac{\sqrt{mn}}{m+\frac{1}{n}}
    • n ≥ 4Q^2
    • m = k + \sqrt{k^2 - 1}, k = (\frac{n}{2Q^2}) - 1
  • Modification: Independent (of Q) gain
    • A{new} = \frac{R{1B}}{R{1A}+R{1B}} A_{old}
    • R1 = R{1A}||R_{1B}
    • R{1A} = R1 \frac{A{old}}{A{new}}
    • R{1B} = \frac{R1}{1-\frac{A{new}}{A{old}}}
• Sallen-Key (KRC) Configuration (High-pass filter)
  • Design equations:
    • H{0HP} = K = 1 + \frac{RB}{R_A}
    • \omega0 = \sqrt{\frac{1}{R1C1R2C_2}}
    • Q = \frac{1}{(1-K)\sqrt{\frac{R2C2}{R1C1}}+\sqrt{\frac{R1C2}{R2C1}}+\sqrt{\frac{R1C1}{R2C2}}}
• Sallen-Key (KRC) Configuration (Band-pass filter)
  • Design equations:
    • H{0BP} = \frac{K}{\frac{1+(1-K)R1}{R3}+(1+\frac{C1}{C2})\frac{R1}{R_2}}
    • \omega0 = \sqrt{\frac{1+\frac{R1}{R3}}{R1C1R2C_2}}
    • Q = \frac{\sqrt{1+\frac{R1}{R3}}}{[1+(1-K)\frac{R1}{R3}]\sqrt{\frac{R2C2}{R1C1}}+\sqrt{\frac{R1C2}{R2C1}}+\sqrt{\frac{R1C1}{R2C2}}}
  • Simplified design equations: Q > \sqrt{2/3}, R1 = R2 = R3 = R, C1 = C_2 = C
    • H_{0BP} = \frac{K}{4-K}
    • \omega_0 = \frac{\sqrt{2}}{RC}
    • Q = \frac{\sqrt{2}}{4-K}
• Sallen-Key (KRC) Configuration (Notch filter)
  • Design equations:
    • H{0NF} = K = 1 + \frac{RB}{R_A}
    • \omega_0 = \frac{1}{RC}
    • Q = \frac{1}{4-2K}
• Multiple Feedback Configuration (Band-pass filter)
  • Design equations:
    • H{0BP} = -\frac{R2/R1}{1+C1/C_2}
    • \omega0 = \sqrt{\frac{1}{R1C1R2C_2}}
    • Q = \sqrt{\frac{R2/R1}{C1/C2}}+\sqrt{\frac{C2}{C1}}
  • Design for |H{0BP}| = H0 < 2Q^2:
    • R{1A} = \frac{Q}{H0\omega_0C}
    • R{1B} = R{1A}(\frac{2Q^2}{H_0} - 1)
• Multiple Feedback Configuration (Low-pass filter)
  • Design equations:
    • H{0LP} = -\frac{R3}{R_1}
    • \omega0 = \sqrt{\frac{1}{R2C1R3C_2}}
    • Q = \sqrt(\frac{C1/C2}{R2R3/R1^2})+\sqrt{\frac{R3}{R2}}+\sqrt{\frac{R2}{R_3}}
  • One approach: C1 = nC2, n ≥ 4Q^2(1 + |H_{0LP}|)
    • R3 = \frac{1 + \sqrt{1-\frac{4Q^2(1+|H{0LP}|)}{n}}}{2\omega0QC2}
    • R1 = \frac{R3}{|H_{0LP}|}
    • R2 = \frac{1}{\omega0^2R3C1C_2}
• 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:
    • \omega0 = \sqrt{\frac{R5}{R4}}\sqrt{\frac{R6}{R7C2}}
    • Q = \frac{(1+R2/R1)\sqrt{R5R6C1/R4R7C2}}{1+R5/R3+R5/R4}
    • H{0HP} = -\frac{R5}{R3}, H{0LP} = -\frac{R4}{R3}, H{0BP} = \frac{1+R2/R1}{1+R3/R4+R3/R_5}
• State Variable Filter Topology
  • Design equations (simplified with mostly equal component): R5 = R4 = R3, R6 = R7 = R, C1 = C_2 = C
    • \omega_0 = \frac{1}{RC}
    • Q = \frac{1}{3}(1 + R2/R1)
    • H{0HP} = -1, H{0LP} = -1, H_{0BP} = Q
• Biquad Filter Topology
  • Design equations:
    • \omega0 = \sqrt{\frac{1}{R4C1R5C_2}}
    • Q = R2\sqrt{\frac{C1}{R4R5C_2}}
    • H{0LP} = \frac{R5}{R1}, H{0BP} = -\frac{R2}{R1}
  • Simplified: R5 = R4 = R, C1 = C2 = C
    • \omega_0 = \frac{1}{RC}
    • Q = \frac{R_2}{R}
    • H{0LP} = \frac{R}{R1}, H{0BP} = -\frac{R2}{R_1}
• Biquad Filter Topology (Notch design)
  • Similar approach to MFB notch filter design
    • Use a BPF
    • Use a summation amplifier
  • Simplified (symmetric notch):
    • \omegaz = \omega0 = \frac{1}{RC}
    • Q = \frac{R_1}{R}
    • H{0N} = -\frac{R5}{R_2}