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.
ζ , Q, magnitude at fo and -3dB fc vs. filter response type.
Figure 3 shows the relationship between the magnitude at fo and the filter ζ . Notice that filter responses with higher ζ have high levels of attenuation at fo , which means that there is more attenuation in the pass band with slower rates of attenuation after fo . Filters with lower ζ exhibit peaking at fo with magnitude levels that are higher than the DC level. For example, a 3dB Chebychev filter has a ζ of 0.3846 and peaking of 2.27dB at fo (the response peaks to 3dB slightly before fo , which is where the response name comes from).
Magnitude change at fo from the DC level vs. ζ .
What you’ll find in many textbooks and active filter design software programs is that coefficients or correction factors are provided to adjust the circuit’s natural frequency so that the response will equal -3dB at the desired fc .
Equation 2 calculates the fo required to achieve the desired magnitude reduction in decibels (x > 0) at fc . Figure 4 displays the values for the correction factor (k) vs. the filter ζ to achieve a -3dB magnitude reduction at fc .
Correction factor (k) for -3dB at fc vs. ζ.
By calculating the proper k, the five circuit responses in Figure 2 can be redesigned to achieve a -3dB reduction at 1kHz. Figure 5 shows the AC transfer function results for the updated circuits. All of the responses now have a -3dB frequency of 1kHz, with the 3dB Chebychev featuring the fastest attenuation rate after 1kHz and the real-pole circuit displaying the slowest attenuation rate after 1kHz.
AC transfer function results for five filter responses with 1kHz-3dB cutoff frequencies.
Table 2 lists the ζ , k and fo for the five filter response types to achieve a -3dB fc of 1kHz.
ζ , k and fo for the filter responses shown in Figure 5.
The fc for Chebychev filters is occasionally defined as the frequency where peaking ends and the magnitude drops back below the DC level. You can accomplish this by using a slightly different equation to calculate k. Table 3 shows the k for a 0dB magnitude at fc = 1kHz for the 0.5dB and 3dB Chebychev filters.
ζ , k and fo for a 0dB Chebychev fc .
In this article, we introduced the transfer equation for a standard second-order active low-pass filter and explained how to achieve different filter response types by varying the filter damping ratio/quality factor. You can use the equations we provided to calculate the natural frequency for the filter and achieve a desired reduction in magnitude at the selected cutoff frequency. Figure 4 displays the natural frequency correction factor required to achieve a -3dB magnitude reduction at the cutoff frequency for different damping ratios.
While this article focused on the Sallen-Key low-pass filter, you can apply the same principles to active high- and band-pass filters, along with other Op Amp circuit topologies such as the multiple feedback topology.