Active and passive analog filters
To reject 40 dB at twice the cutoff frequency, a Butterworth needs seven poles and a 0.5 dB Chebyshev, five. The ripple in the passband buys steepness.
01The filter template
A filter is specified with a template: in the passband, up to \( \omega_p \), the attenuation cannot exceed \( A_p \); in the stopband, from \( \omega_s \) on, it has to be at least \( A_s \). Between the two lies the transition band. The narrower the transition and the more demanding \( A_p \) and \( A_s \), the higher the order needed: more poles, more components. Highpass, bandpass and bandstop filters are designed from a normalized lowpass with a frequency transformation, so everything starts with the lowpass. The basic filters are presented in the technician program, in filters.
02The approximations
The ideal filter —gain 1 up to the cutoff and 0 after it— cannot be built. An approximation is a realizable transfer function that comes close to it in a particular way, and each one chooses what to sacrifice:
| Approximation | Passband | Transition | Step response |
|---|---|---|---|
| Butterworth | Maximally flat | Moderate | Moderate overshoot |
| Chebyshev | With ripple of \( A_p \) | Steeper for the same order | Greater overshoot and ringing |
| Inverse Chebyshev | Flat | Steep, with zeros in the stopband | Moderate |
| Elliptic (Cauer) | With ripple | The steepest possible | The worst |
| Bessel | Rolls off gently | Very slow | Almost no overshoot: constant group delay |
The minimum order comes from the template:
03Passive ladder filters
The classic passive filter is an LC ladder between the generator resistance and the load resistance, with series inductors and shunt capacitors: the Cauer form from network functions. Its values are tabulated in normalized form, for 1 Ω and 1 rad/s; for Butterworth they are \( g_k = 2\sin\frac{(2k - 1)\pi}{2n} \): a third-order filter has 1 H, 2 F and 1 H. They are denormalized by dividing the inductances and capacitances by \( \omega_c \) and scaling by the impedance. Doubly terminated ladders have a notable advantage: their response is very insensitive to component tolerances.
04Active filters
At low frequencies inductors are large, expensive and far from ideal. An active filter replaces the ladder with a cascade of second-order sections using operational amplifiers, each one with a pair of poles. The most widely used is the Sallen-Key: with unity gain and equal resistors \( R \), the capacitors set the frequency and the quality factor of the section:
The sections are ordered with increasing Q, so that the high-Q ones do not saturate on signals that the next section was going to attenuate. The op amp has to have a gain-bandwidth product well above the \( Q \cdot f_0 \) of each section; otherwise, the response is distorted.
The response of a normalized lowpass of the chosen type and order, with the specification template shaded. Below, the minimum order to meet it and the unity-gain Sallen-Key sections for a 1 kHz cutoff with 10 kΩ resistors.
05Where it appears in electronics
| Application | Typical filter |
|---|---|
| Anti-aliasing before a converter | Active Butterworth or Bessel of order 2 to 8 |
| Transmitter output stage | LC ladder, Chebyshev or elliptic, for the harmonics |
| Pulse measurement or high-quality audio | Bessel, for its step response without overshoot |
| Loudspeaker crossover | Linkwitz-Riley: two Butterworth filters in cascade |
06In the lab
Build the two Sallen-Key sections from the lab for 1 kHz, measure the frequency response and compare it with the theoretical one.
Compare the square-wave response of a Butterworth and a 1 dB Chebyshev of the same order.
Denormalize a fifth-order LC Butterworth for 10 MHz and 50 Ω, build it and measure it with a network analyzer.
07Common mistakes
- Mixing up the normalizations. Butterworth is normalized to −3 dB; Chebyshev, to the edge of the ripple.
- Using a slow op amp in a high-Q section. The peak shifts and grows.
- Ordering the sections wrongly. The high-Q ones go last.
- Choosing by the frequency response alone. If the waveform matters, you have to look at the group delay.
08Self-assessment
How much does a fourth-order Butterworth attenuate at twice its cutoff frequency?
\( 10\log(1 + 2^8) \approx 24.1 \) dB.
What Q values do the sections of a fourth-order Butterworth have?
0.541 and 1.307, both with \( \omega_0 = 1 \).
What capacitors does a unity-gain Sallen-Key with Q = 0.707, 1 kHz and 10 kΩ use?
\( C_1 = 2 \cdot 0.707 / (2\pi \cdot 1000 \cdot 10^4) \approx 22.5 \) nF and \( C_2 \approx 11.3 \) nF.
Why does a Chebyshev need a lower order than a Butterworth?
Because it accepts ripple in the passband and in exchange falls faster in the transition.
09Further reading
- Arthur B. Williams and Fred J. Taylor. Electronic Filter Design Handbook. 4th ed., McGraw-Hill, 2006. The tables of poles, sections and normalized ladders for all the approximations.
- Rolf Schaumann, Haiqiao Xiao and Mac E. Van Valkenburg. Design of Analog Filters. 2nd ed., Oxford University Press, 2009. The design of active and passive filters with the complete theory.
- Texas Instruments. Op Amps for Everyone, active filters chapter. Texas Instruments, various editions. The practical design of Sallen-Key and multiple-feedback filters, with coefficient tables.