Active Filters#

Op-amp-based filter designs that overcome the limitations of passive RC filters: they can provide gain, don’t load the source, have low output impedance, and can achieve higher Q and sharper roll-off without inductors. The tradeoff is complexity, power consumption, and new failure modes (op-amp bandwidth, noise, and stability).

Why Active Filters?#

Passive RC filters have fundamental limitations:

  • No gain — A passive filter can only attenuate. If a circuit needs to filter and amplify, it requires two stages
  • Loading — Cascading passive stages requires impedance management or buffers
  • No inductorless resonance — Getting a sharp (high-Q) band-pass with only R and C requires active feedback. Passive high-Q requires inductors, which are large, expensive, and have parasitic resistance
  • Limited Q — Passive RLC filters’ Q is limited by component losses (inductor DCR, capacitor ESR)

Active filters solve all of these by using op-amp feedback to synthesize the desired transfer function.

Common Topologies#

Sallen-Key#

The most widely used active filter topology. Uses a single op-amp in a non-inverting configuration with frequency-dependent feedback.

  • Provides second-order (two-pole) response per op-amp
  • Low component sensitivity for Butterworth (maximally flat) response
  • Easy to design — many online calculators and cookbook formulas
  • Gain is typically unity or low (the op-amp acts more as a buffer than an amplifier)

Limitation: High-Q designs (Q > 5) become sensitive to component tolerances. Small changes in R or C values significantly affect the response shape.

Multiple Feedback (MFB)#

An inverting topology with two feedback paths (one through a resistor, one through a capacitor).

  • Also second-order per op-amp
  • Inverting (180-degree phase shift)
  • Lower sensitivity to op-amp gain-bandwidth than Sallen-Key at higher frequencies
  • Better suited for higher-Q designs than Sallen-Key
  • More complex design equations

State Variable#

Uses three op-amps to produce simultaneous low-pass, band-pass, and high-pass outputs from the same circuit.

  • Second-order per section
  • Q and frequency can be adjusted independently
  • Low sensitivity to component tolerances
  • Higher component count (three op-amps per second-order section)
  • The go-to topology when tunable Q or frequency is needed

Biquad#

Similar to state variable but with a different feedback structure. Also produces simultaneous outputs with independently adjustable parameters.

Filter Response Types#

The transfer function mathematics determines the shape of the filter response. The most common types:

Butterworth (Maximally Flat):

  • Flattest possible passband — no ripple
  • Monotonic roll-off
  • -3 dB at the cutoff frequency
  • Moderate phase nonlinearity
  • The default choice when there are no specific requirements

Chebyshev (Type I):

  • Steeper roll-off than Butterworth for the same order
  • Ripple in the passband (designer-specified — 0.5 dB and 1 dB are common)
  • Sharper transition band
  • Worse phase nonlinearity
  • Use when a sharper cutoff is needed and passband ripple is tolerable

Bessel (Maximally Flat Phase):

  • Best phase linearity (preserves waveform shape)
  • Slowest roll-off of the three
  • Best step response (minimal overshoot and ringing)
  • Use for pulse/digital signals where waveform fidelity matters

Design Process#

  1. Define requirements — Cutoff frequency, roll-off rate (which sets the filter order), passband ripple tolerance, stopband attenuation
  2. Choose response type — Butterworth for general use, Chebyshev for sharp cutoff, Bessel for pulse fidelity
  3. Choose topology — Sallen-Key for simplicity, MFB for higher Q, state variable for tunability
  4. Calculate component values — Use standard design tables (coefficient tables for each response type) and scale for the desired frequency and impedance level
  5. Verify with simulation — Component tolerances and op-amp limitations can shift the response. Simulate before building
  6. Build and measure — Compare measured response to design and simulation. Adjust if needed

Stability and Component Sensitivity#

Active filters use positive feedback (indirectly, through the filter’s frequency-dependent network) to create resonance. This makes them potentially unstable:

  • High-Q sections are the most sensitive — A small component change can push Q toward infinity (oscillation) or collapse it (overdamped response). This is why Q > 10 active filters are difficult in practice
  • Op-amp GBW matters — The op-amp must have enough bandwidth that its gain is still high at the filter’s cutoff frequency. A rule of thumb: GBW should be at least 10x the filter’s highest cutoff frequency for Sallen-Key, and more for high-Q designs
  • Component matching — For higher-order filters (cascaded second-order sections), the Q and frequency of each section must be accurate. Use 1% or better resistors and 5% or better capacitors. For critical applications, use 1% capacitors (C0G/NP0)

Tips#

  • Select an op-amp with GBW at least 10× the filter’s highest cutoff frequency for Sallen-Key, more for high-Q designs
  • Use C0G/NP0 capacitors for timing-critical filter sections to avoid voltage-dependent capacitance
  • Simulate the filter before building — component tolerances and op-amp limitations can shift the response significantly
  • For Q > 5, consider state-variable topology for lower component sensitivity

Caveats#

  • Op-amp bandwidth limits the filter — Above the op-amp’s useful bandwidth, the filter response deviates from the design. The filter may even have gain peaks at frequencies where the op-amp’s phase shift interacts with the feedback network
  • Noise floor — Active filters add noise from the op-amp. The noise appears at the output, potentially limiting dynamic range. Low-noise op-amps help, but the fundamental thermal noise of the filter resistors sets a floor
  • Power supply rejection — Supply noise can couple into the filter output, especially at frequencies near the filter’s passband. Good decoupling is essential
  • DC offset — Op-amp offset voltage appears at the output, potentially amplified by the filter’s DC gain. For band-pass and high-pass filters, this is naturally blocked. For low-pass filters, it passes through
  • Capacitor selection matters — C0G/NP0 capacitors are preferred for active filters because their capacitance doesn’t change with voltage, temperature, or time. X7R introduces nonlinearity (voltage-dependent capacitance means signal-dependent filter characteristics — a subtle form of distortion)

In Practice#

  • A filter that oscillates has Q that’s too high for the component tolerances — reduce Q or use tighter-tolerance parts
  • Gain peaking near the cutoff frequency suggests underdamped poles (Q > 0.707) — check component values against design
  • Distortion that varies with signal level may indicate voltage-dependent capacitance — switch to C0G/NP0 capacitors
  • Response that deviates from simulation at high frequencies indicates the op-amp GBW is limiting — use a faster op-amp
Backward RC & RL Filters Frequency Response Forward