Active filters are the superheroes of signal processing. They use op-amps to boost performance, allowing for steeper roll-offs and higher Q factors than passive filters. Plus, they're easier to tweak and don't load down your circuit.

This section dives into different active filter designs like Sallen-Key and multiple feedback. You'll learn how to choose the right topology for your needs, whether you're building a low-pass filter for audio or a band-pass filter for wireless comms.

Filter Types

Low-Pass and High-Pass Filters

• Low-pass filter allows frequencies below cutoff frequency to pass through
  • Attenuates higher frequencies
  • Applications include audio systems, removing high-frequency noise
  • Transfer function: $H(s) = \frac{\omega_c}{s + \omega_c}$
• High-pass filter permits frequencies above cutoff frequency to pass
  • Blocks lower frequencies
  • Used in audio crossovers, AC coupling circuits
  • Transfer function: $H(s) = \frac{s}{s + \omega_c}$
• Cutoff frequency ($f_c$) determines filter behavior
  • Calculated using resistor and capacitor values: $f_c = \frac{1}{2\pi RC}$
• Gain response varies with frequency
  • Low-pass: -20 dB/decade slope above cutoff
  • High-pass: -20 dB/decade slope below cutoff

Band-Pass and Band-Stop Filters

• Band-pass filter allows a specific range of frequencies to pass
  • Combines low-pass and high-pass characteristics
  • Used in wireless communications, audio equalization
  • Transfer function: $H(s) = \frac{As}{s^2 + (\omega_0/Q)s + \omega_0^2}$
• Band-stop filter (notch filter) attenuates a specific frequency range
  • Blocks unwanted frequencies while allowing others to pass
  • Applications include noise reduction, eliminating power line interference
  • Transfer function: $H(s) = \frac{s^2 + \omega_0^2}{s^2 + (\omega_0/Q)s + \omega_0^2}$
• Center frequency ($f_0$) and bandwidth define filter response
  • Center frequency: $f_0 = \frac{1}{2\pi \sqrt{LC}}$
  • Bandwidth: $BW = \frac{f_0}{Q}$
• Q factor influences filter selectivity
  • Higher Q results in narrower bandwidth and steeper roll-off

Filter Characteristics

Filter Response Types

• Butterworth filter provides maximally flat passband response
  • Smooth roll-off in stopband
  • Moderate selectivity and phase response
  • Transfer function: $|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2n}}$
• Chebyshev filter offers steeper roll-off but with passband ripple
  • Type I: Ripple in passband, flat stopband
  • Type II: Flat passband, ripple in stopband
  • Improved selectivity compared to Butterworth
  • Transfer function: $|H(j\omega)|^2 = \frac{1}{1 + \epsilon^2 T_n^2(\omega/\omega_c)}$

Filter Design Parameters

• Filter order determines roll-off rate and stopband attenuation
  • Higher order increases slope steepness (n -20 dB/decade)
  • Affects component count and circuit complexity
• Cutoff frequency marks -3 dB point in frequency response
  • Defines boundary between passband and stopband
  • Calculated using component values in active filter circuits
• Q factor measures filter's selectivity and bandwidth
  • Higher Q results in sharper resonance peak
  • Influences filter's transient response and ringing
  • Calculated as: $Q = \frac{f_0}{BW}$

Active Filter Topologies

Sallen-Key and Multiple Feedback Topologies

• Sallen-Key topology offers simple design and low component count
  • Uses positive feedback for improved performance
  • Can implement low-pass, high-pass, and band-pass filters
  • Transfer function (low-pass): $H(s) = \frac{K\omega_0^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}$
• Multiple feedback topology provides high Q factors and gain
  • Uses negative feedback for stability
  • Suitable for band-pass and notch filter designs
  • Transfer function (band-pass): $H(s) = \frac{-sR_2C_1}{1 + s(C_1R_1 + C_1R_2 + C_2R_2) + s^2R_1R_2C_1C_2}$
• Both topologies utilize operational amplifiers as active elements
  • Op-amps provide gain and impedance buffering
  • Enable higher Q factors compared to passive filters

Advanced Filter Structures

• State variable filter offers simultaneous low-pass, high-pass, and band-pass outputs
  • Uses multiple op-amps for increased flexibility
  • Allows independent control of Q and cutoff frequency
  • Transfer function (band-pass): $H(s) = \frac{K\omega_0s}{s^2 + \omega_0s/Q + \omega_0^2}$
• Biquad filter implements second-order transfer functions
  • Cascadable for higher-order filters
  • Provides low sensitivity to component variations
  • Transfer function: $H(s) = \frac{b_2s^2 + b_1s + b_0}{s^2 + a_1s + a_0}$
• Both structures offer improved performance and versatility
  • Used in audio processing, instrumentation, and communications systems
  • Allow realization of complex filter responses