How Do You Design an Active Filter with an Op-Amp?

Detailed Explanation

An active filter is a frequency-selective circuit built from op-amps, resistors, and capacitors. By incorporating an op-amp's gain into the filter topology, it achieves sharp roll-off and defined passband behaviour that a passive RC network cannot match without inductors. Active filters are used in:

• Anti-aliasing before an ADC (band-limiting the signal to below Nyquist)
• Audio equalisation and crossover circuits
• Sensor signal conditioning (removing out-of-band noise before amplification)
• Frequency synthesis and waveform generation — a reconstruction filter on a DAC output smooths the staircase waveform
• Notch filters for mains interference rejection (50 Hz in Australia)

When the signal originates from a differential sensor (Wheatstone bridge, thermocouple, load cell), an instrumentation amplifier typically provides the initial differential gain before the active filter band-limits the amplified signal.

First-Order Passive RC Review

A single resistor-capacitor combination forms a first-order low-pass filter with corner frequency fc = 1/(2πRC). The response rolls off at −20 dB/decade above fc (−6 dB/octave). This is often insufficient: at one decade above fc, the signal is attenuated only 20 dB; at two decades, 40 dB. For noise or alias rejection requiring 60 dB or more, first-order is rarely enough.

Active filters allow second- and higher-order responses to be built from only Rs and Cs, achieving −40, −60, or −80 dB/decade without inductors.

Second-Order Low-Pass: Key Parameters

A second-order filter has two poles and a transfer function characterised by:

• Corner (pole) frequency, fc: The −3 dB point (Butterworth) or the frequency at which the filter transitions from passband to stopband.
• Quality factor, Q (or damping ζ = 1/(2Q)): Controls the shape of the response near fc. Butterworth: Q = 0.707, maximally flat passband, −3 dB exactly at fc. Chebyshev: Q > 0.707, passband ripple in exchange for steeper initial rolloff. Bessel: Q < 0.707, maximally flat group delay (linear phase), gentler rolloff but best step response.

For most ADC anti-aliasing work, Butterworth (Q = 0.707) is the default: no passband ripple and predictable attenuation.

Sallen-Key Low-Pass Topology

The Sallen-Key is the most common active filter topology for low to moderate Q values (Q up to ~2):

Vin ──R1──┬──R2──┬── (+) ──────── Vout
          C1    │            │
         GND   C2           │
               │           (−)
              GND    Rf ────┤
                    │      │
                   Rg──GND

For the unity-gain Sallen-Key (voltage follower at the op-amp output, no Rf or Rg):

• Corner frequency: fc = 1 / (2π√(R1 × R2 × C1 × C2))
• Q = √(R1 × R2 × C1 × C2) / (C2 × (R1 + R2))

Equal-component design (R1 = R2 = R, C1 = C2 = C):

• fc = 1/(2πRC)
• Q = 0.5

Q = 0.5 is below Butterworth (Q = 0.707), giving a Critically Damped or Bessel-like response with no peaking. To achieve Butterworth Q = 0.707 with equal Rs, use C1 = 2C2. Practical equal-component values are easy to procure and the formula simplifies to fc = 1/(2πRC).

Sallen-Key with gain: Add a non-inverting gain stage (Rf/Rg resistors) at the op-amp to achieve passband gain greater than unity and simultaneously raise Q. Gain K sets Q = 1/(3 − K) for equal-component design — the maximum useful gain is K < 3 before the circuit becomes unstable.

Multiple Feedback (MFB) Low-Pass Topology

The MFB topology inverts the signal and can achieve higher Q values than Sallen-Key without risk of instability:

Vin ──R1──┬──R2──(−)─── Vout
          │         │
          C1       Cf
          │         │
         R3──(+)   │
              │    │
             GND───┘

Corner frequency and Q are set by three resistors and two capacitors. The MFB is preferred when:

• Q > 1 is required (e.g. Chebyshev approximation)
• Signal inversion is acceptable
• Higher-order filters are built by cascading two-pole MFB stages

Filter Approximations

Butterworth (maximally flat): No passband ripple, −3 dB at fc, rolloff is −20n dB/decade for an nth-order filter. The most common choice.

Chebyshev type I: Passband ripple (0.5 dB or 1 dB typical), steeper rolloff than Butterworth for the same order, useful when stopband attenuation matters more than passband flatness.

Bessel (Thomson-Bessel): Maximally flat group delay (constant time delay across the passband), meaning all frequencies are delayed by the same amount — the filter does not distort the phase of the signal. Important in data communications and any application where pulse shape must be preserved. Rolloff is gentler than Butterworth for the same order.

Practical Anti-Aliasing Filter Design

For an ADC sampling at fs, the Nyquist frequency is fs/2. The anti-aliasing filter must attenuate signals above fs/2 sufficiently to prevent aliases that would corrupt the in-band measurement.

Example: 12-bit ADC, fs = 10 kSPS, signal bandwidth = 1 kHz, target alias attenuation = 72 dB (12-bit noise floor).

1. Choose fc = 1 kHz (preserves the full signal bandwidth to −3 dB at 1 kHz)
2. At Nyquist (5 kHz), need 72 dB attenuation: 5 kHz / 1 kHz = 5×, so 14 dB/decade (first order), need about 72/14 = 5 decades — clearly a first-order filter is far from enough.
3. A fourth-order Butterworth at fc = 1 kHz provides −80 dB at 5 kHz (1 decade above fc gives 80 dB from a 4th-order filter). This meets the requirement.
4. Build as two cascaded second-order Sallen-Key sections.

In practice, ADC manufacturers specify a recommended input filter in the datasheet. This filter provides the Riso that prevents the ADC's input switching current from coupling back into the amplifier driving it, in addition to alias filtering — both are often needed. See ADC fundamentals for the Nyquist relationship and the op-amp capacitive load note for a real example of how the anti-aliasing filter interacts with the driving op-amp.

Filter Design Software

Designing active filters from scratch is tedious. Use free tools:

• Texas Instruments FilterPro / WEBENCH Filter Designer: Enters target fc, order, and approximation type; generates resistor/capacitor values and SPICE simulation.
• Analog Devices ADIsimFilter: Similar interactive tool.
• Microchip FilterLab: Focused on Sallen-Key and MFB topologies with standard component values.

For preliminary calculations, use online Butterworth filter calculators and cross-check with the design equations.

For analog front-end design including anti-aliasing filter integration as part of a complete signal chain, Zeus Design's electronics engineering team covers circuit design through to PCB layout.

Design Considerations

• Op-amp GBW must exceed filter frequency: The op-amp's closed-loop bandwidth must comfortably exceed the filter corner frequency at its operating gain. As a rule of thumb, choose GBW ≥ 100 × fc at unity gain. For a 10 kHz anti-aliasing filter, a 1 MHz GBW op-amp is marginal; use ≥ 10 MHz.
• Capacitor dielectric matters: Use C0G (NP0) capacitors for precision filters — they have stable capacitance vs temperature and voltage. X5R/X7R ceramics change capacitance with DC bias and temperature, shifting the corner frequency.
• Use standard E96 resistor values: Design tools typically provide ideal resistor values; round to the nearest E96 value and accept the small frequency shift. A 1% deviation in a resistor shifts fc by 0.5%.
• Stability with capacitive load: Op-amps in active filter configurations can be sensitive to capacitive loading at the output. If the filter drives a long cable or high-capacitance load, add a small series resistor (10–100 Ω) at the output after the feedback takeoff point.

Common Mistakes

• Placing the anti-aliasing filter inside the feedback loop: If the feedback resistor of a preceding op-amp stage connects after the filter capacitor, the filter capacitance degrades the op-amp's phase margin and can cause oscillation. The feedback takeoff point should always be at the op-amp output pin, not after a passive filter element.
• Ignoring the Riso requirement for ADC inputs: The ADC's internal sample-and-hold switch charges and discharges an internal capacitor at the sampling rate, creating current spikes that appear on the input pin. These can couple back through a low-impedance source and cause measurement errors. Always include the ADC manufacturer's recommended input RC network — the series resistor serves as both Riso and part of the anti-aliasing filter.
• Cascading high-Q stages without checking op-amp bandwidth: Each cascaded stage requires adequate op-amp GBW. A 4th-order filter using two op-amps with marginal GBW will shift its corner frequency and change its response shape from the design target.