Editor’s note: I am proud to bring you this article authored by both Collin Wells, Applications Manager, General Purpose amplifier products, Texas Instruments and Gustaf Falk Olson, Field Applications Engineer, Texas Instruments
In this Signal Chain Basics, we’ll discuss the design of different operational amplifier (op amp) active filter response types using basic circuit-transfer function equations. Many filter design programs, including Texas Instruments FilterPro™ software or WEBENCH® Active Filter Designer, assist in the design of op amp active filters. However, this article provides a few details missing from other resources that we thought would be helpful to engineers who like to design circuits or double-check their circuit designs by hand.
We will use the second-order Sallen-Key low-pass filter as the basis for this article, although you can apply the principles to other filter types and topologies. The Sallen-Key circuit topology creates a noninverting response with low sensitivity to component values. Figure 1 shows an example second-order Sallen-Key low-pass filter.
1kHz Butterworth Sallen-Key low-pass filter.
Equation 1 shows the transfer function (Vo/Vin) for the circuit in Figure 1. By arranging the transfer function equation in the standard form for a second-order low-pass filter, you can easily identify the damping ratio ( ζ ), quality factor (Q) and natural frequency (ωo in radians, fo in Hz) of the circuit.
You can achieve different filter response types for the second-order filter by varying the ζ or Q of the circuit. With the component values shown in Figure 1, the circuit has a Butterworth response with a -3dB cutoff frequency (fc) of 1kHz. A Butterworth response has a ζ and Q both equal to 0.707 and offers maximal flatness in the pass band, with a good rate of attenuation after fc. Other popular response types include Bessel and 0.5 and 3dB Chebychev (Tschebyscheff). The Bessel response has a lower Q and a higher ζ , which results in a linear phase response with transient behavior featuring minimal overshoot and ringing at the expense of an earlier roll-off in the pass band and a slower rate of attenuation after fc. The Chebychev responses have higher Qs and lower ζ s, resulting in peaking in the pass band (0.5dB or 3dB) with higher transient overshoot and ringing, but faster rates of attenuation after fc.
Figure 2 shows the results of five different filter response types, each with a natural frequency (fo) of 1kHz. Figure 2 also shows results for a circuit with real poles ( ζ = 1) for reference, in order to compare what a passive resistor-capacitor (RC) filter response would look like with two real poles at the same frequency.
AC transfer function results for five filter responses with 1kHz (fo).
The response types have differences in pass-band flatness and different rates of attenuation after the 1kHz natural frequency. Notice that the fo of 1kHz does not correspond to the -3dB frequency for any of the responses other than the Butterworth response. This is often a point of confusion among active filter designers because they assume that fo will equal the -3dB fc. Table 1 lists the ζ , Q, magnitude at fo (1kHz) and -3dB fc for the five filter response types in Figure 2.
, Q, magnitude at fo and -3dB fc vs. filter response type.