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: $H_{LPF}(jω) = \frac{1}{1 - (ω/ω0)^2 + (jω/ω0)/Q}$
    • $|H<em>{LP}|</em>{max} = Q \sqrt{1 - 1/4Q^2}$
  • High-pass filter response: $H_{HPF}(jω) = \frac{-(ω/ω0)^2}{1 - (ω/ω0)^2 + (jω/ω0)/Q}$
  • Band-pass filter response: $H_{BPF}(jω) = \frac{(jω/ω0)/Q}{1 - (ω/ω0)^2 + (jω/ω0)/Q}$
    • $\omega<em>L = \omega</em>0 \sqrt{1 + \frac{1}{4Q^2}} - \frac{1}{2Q}$, $\omega<em>H = \omega</em>0 \sqrt{1 + \frac{1}{4Q^2}} + \frac{1}{2Q}$
    • $\omega<em>0 = \sqrt{\omega</em>L\omega<em>H}$, $Q = \frac{\omega</em>0}{BW}$

Second Order Filter Configurations

• Sallen-Key (KRC) Configuration (Low-pass filter)
  • Transfer function: $H(jω) = K \frac{1}{1 - ω^2R<em>1C</em>1R<em>2C</em>2 + jω [(1 - K) R<em>1C</em>1 + R<em>1C</em>2 + R<em>2C</em>2]}$
  • Standard design equations:
    • $H<em>{0LP} = K = 1 + \frac{R</em>B}{R_A}$
    • $\omega<em>0 = \sqrt{\frac{1}{R</em>1C<em>1R</em>2C_2}}$
    • $Q = \frac{1}{(1-K)\sqrt{R<em>1C</em>1/R<em>2C</em>2}+\sqrt{R<em>1C</em>2/R<em>2C</em>1}+\sqrt{R<em>2C</em>2/R<em>1C</em>1}}$
  • Simplification: Equal components ($R = R<em>1 = R</em>2$, $C = C<em>1 = C</em>2$)
    • $H<em>{0LP} = K = 1 + \frac{R</em>B}{R_A}$
    • $\omega_0 = \frac{1}{RC}$
    • $Q = \frac{1}{(3-K)}$
  • Simplification: Unity gain ($H_{0LP} = 1 V/V$)
    • $R<em>2 = R, R</em>1 = mR, C<em>2 = C, C</em>1 = nC$
    • $\omega_0 = \sqrt{\frac{1}{mnR^2C^2}} = \frac{1}{RC\sqrt{mn}}$
    • $Q = \frac{\sqrt{mn}}{m+\frac{1}{n}}$
    • $n ≥ 4Q^2$
    • $m = k + \sqrt{k^2 - 1}, k = (\frac{n}{2Q^2}) - 1$
  • Modification: Independent (of Q) gain
    • $A<em>{new} = \frac{R</em>{1B}}{R<em>{1A}+R</em>{1B}} A_{old}$
    • $R<em>1 = R</em>{1A}||R_{1B}$
    • $R<em>{1A} = R</em>1 \frac{A<em>{old}}{A</em>{new}}$
    • $R<em>{1B} = \frac{R</em>1}{1-\frac{A<em>{new}}{A</em>{old}}}$
• Sallen-Key (KRC) Configuration (High-pass filter)
  • Design equations:
    • $H<em>{0HP} = K = 1 + \frac{R</em>B}{R_A}$
    • $\omega<em>0 = \sqrt{\frac{1}{R</em>1C<em>1R</em>2C_2}}$
    • $Q = \frac{1}{(1-K)\sqrt{\frac{R<em>2C</em>2}{R<em>1C</em>1}}+\sqrt{\frac{R<em>1C</em>2}{R<em>2C</em>1}}+\sqrt{\frac{R<em>1C</em>1}{R<em>2C</em>2}}}$
• Sallen-Key (KRC) Configuration (Band-pass filter)
  • Design equations:
    • $H<em>{0BP} = \frac{K}{\frac{1+(1-K)R</em>1}{R<em>3}+(1+\frac{C</em>1}{C<em>2})\frac{R</em>1}{R_2}}$
    • $\omega<em>0 = \sqrt{\frac{1+\frac{R</em>1}{R<em>3}}{R</em>1C<em>1R</em>2C_2}}$
    • $Q = \frac{\sqrt{1+\frac{R<em>1}{R</em>3}}}{[1+(1-K)\frac{R<em>1}{R</em>3}]\sqrt{\frac{R<em>2C</em>2}{R<em>1C</em>1}}+\sqrt{\frac{R<em>1C</em>2}{R<em>2C</em>1}}+\sqrt{\frac{R<em>1C</em>1}{R<em>2C</em>2}}}$
  • Simplified design equations: $Q > \sqrt{2/3}, R<em>1 = R</em>2 = R<em>3 = R, C</em>1 = C_2 = C$
    • $H_{0BP} = \frac{K}{4-K}$
    • $\omega_0 = \frac{\sqrt{2}}{RC}$
    • $Q = \frac{\sqrt{2}}{4-K}$
• Sallen-Key (KRC) Configuration (Notch filter)
  • Design equations:
    • $H<em>{0NF} = K = 1 + \frac{R</em>B}{R_A}$
    • $\omega_0 = \frac{1}{RC}$
    • $Q = \frac{1}{4-2K}$
• Multiple Feedback Configuration (Band-pass filter)
  • Design equations:
    • $H<em>{0BP} = -\frac{R</em>2/R<em>1}{1+C</em>1/C_2}$
    • $\omega<em>0 = \sqrt{\frac{1}{R</em>1C<em>1R</em>2C_2}}$
    • $Q = \sqrt{\frac{R<em>2/R</em>1}{C<em>1/C</em>2}}+\sqrt{\frac{C<em>2}{C</em>1}}$
  • Design for $|H<em>{0BP}| = H</em>0 < 2Q^2$:
    • $R<em>{1A} = \frac{Q}{H</em>0\omega_0C}$
    • $R<em>{1B} = R</em>{1A}(\frac{2Q^2}{H_0} - 1)$
• Multiple Feedback Configuration (Low-pass filter)
  • Design equations:
    • $H<em>{0LP} = -\frac{R</em>3}{R_1}$
    • $\omega<em>0 = \sqrt{\frac{1}{R</em>2C<em>1R</em>3C_2}}$
    • $Q = \sqrt(\frac{C<em>1/C</em>2}{R<em>2R</em>3/R<em>1^2})+\sqrt{\frac{R</em>3}{R<em>2}}+\sqrt{\frac{R</em>2}{R_3}}$
  • One approach: $C<em>1 = nC</em>2, n ≥ 4Q^2(1 + |H_{0LP}|)$
    • $R<em>3 = \frac{1 + \sqrt{1-\frac{4Q^2(1+|H</em>{0LP}|)}{n}}}{2\omega<em>0QC</em>2}$
    • $R<em>1 = \frac{R</em>3}{|H_{0LP}|}$
    • $R<em>2 = \frac{1}{\omega</em>0^2R<em>3C</em>1C_2}$
• 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:
    • $\omega<em>0 = \sqrt{\frac{R</em>5}{R<em>4}}\sqrt{\frac{R</em>6}{R<em>7C</em>2}}$
    • $Q = \frac{(1+R<em>2/R</em>1)\sqrt{R<em>5R</em>6C<em>1/R</em>4R<em>7C</em>2}}{1+R<em>5/R</em>3+R<em>5/R</em>4}$
    • $H<em>{0HP} = -\frac{R</em>5}{R<em>3}, H</em>{0LP} = -\frac{R<em>4}{R</em>3}, H<em>{0BP} = \frac{1+R</em>2/R<em>1}{1+R</em>3/R<em>4+R</em>3/R_5}$
• State Variable Filter Topology
  • Design equations (simplified with mostly equal component): $R<em>5 = R</em>4 = R<em>3$, $R</em>6 = R<em>7 = R$, $C</em>1 = C_2 = C$
    • $\omega_0 = \frac{1}{RC}$
    • $Q = \frac{1}{3}(1 + R<em>2/R</em>1)$
    • $H<em>{0HP} = -1, H</em>{0LP} = -1, H_{0BP} = Q$
• Biquad Filter Topology
  • Design equations:
    • $\omega<em>0 = \sqrt{\frac{1}{R</em>4C<em>1R</em>5C_2}}$
    • $Q = R<em>2\sqrt{\frac{C</em>1}{R<em>4R</em>5C_2}}$
    • $H<em>{0LP} = \frac{R</em>5}{R<em>1}, H</em>{0BP} = -\frac{R<em>2}{R</em>1}$
  • Simplified: $R<em>5 = R</em>4 = R, C<em>1 = C</em>2 = C$
    • $\omega_0 = \frac{1}{RC}$
    • $Q = \frac{R_2}{R}$
    • $H<em>{0LP} = \frac{R}{R</em>1}, H<em>{0BP} = -\frac{R</em>2}{R_1}$
• Biquad Filter Topology (Notch design)
  • Similar approach to MFB notch filter design
    • Use a BPF
    • Use a summation amplifier
  • Simplified (symmetric notch):
    • $\omega<em>z = \omega</em>0 = \frac{1}{RC}$
    • $Q = \frac{R_1}{R}$
    • $H<em>{0N} = -\frac{R</em>5}{R_2}$