# Signal Chain Basics #126: How to design active filters with different response types using circuit-transfer function equations

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.

Figure 1

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.

Figure 2

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.

Table 1

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).

Figure 3

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 .

Figure 4

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.

Figure 5

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.

Table 2

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.

Table 3

ζ , k and fo for a 0dB Chebychev fc .

Conclusion

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.

References

1. Chapter 16, Active Filter Design Techniques.” Excerpted from “Op Amps for Everyone.” Texas Instruments Literature No. SLOA088.
2. Paul Horowitz and Winfield Hill. “The Art of Electronics, 3rd Edition.” Cambridge University Press, 2015.

## 6 comments on “Signal Chain Basics #126: How to design active filters with different response types using circuit-transfer function equations”

1. silicon_researcher
August 2, 2017

In general, the damping ratio changes when the natural frequency (f0 ) is changed. This is clearly seen in the equations for sallen key filter (both are dependent variable of the same independent variables, namely R1 , R2 , C1 , C2 ). In tables 1 & 2 we see that the damping ratios are identical before and after application of correction factor k. In certain restricted cases of choosing cap/resistor values, I can see that one can make them identical. For example choosing Ci (new) = k*Ci (old), i = 1, 2 and thus f0 (new) = 1/k *f0 (old) but damping factor remains unchanged. Or are the damping ratios in table 2  values before correction? Can the authors clarify.

2. sfierro53
August 3, 2017

I have read many articles on how to design active feedback filters which concentrate on the filter parameters and component values. I however have not seen one which explains the neccessary GBW product specification that will be required of the op amp. Filter pro will state the GBW specification but no explanation is given. For instance for a 2 pole 1MHz butterworth LPF with unity gain the GBW product called for is about 84MHz. This is for sallen key or MFB topologies. Can you explain why or give a link to an app note which does?

3. CollinWells
August 3, 2017

Thanks for your comments!  There is a nice explanation of the op amp gain-bandwidth requirements in Chapter 16.8.4 of the popular “Op Amps For Everyone” publication.  Basically the equations define that the op amp should have an open-loop gain magnitude at least 40dB (100V/V) larger than the peak gain (Q) of the filter which will keep the gain error <1%.

4. Tucson_Mike
August 22, 2017

To continue this required GBP in an active filter discussion, essentially it becomes a rule of thumb by whomever is setting up the tool I suspect. Some criteria of fit is required. I tend to target a bit lower as once you start runnning monte-carlos with real C tolerances, you find out quick that is the dominant error source and you can use a lower target GBP without impacting the spread too much.  Some details –

1. Long time ago, in a BurrBrown app note supporting Filterpro, I belive I read Filterpro(which is what Collin is using probably) uses a min GBP = 100*Fo*Ko*(Q^2) and most other tools use this same form with a different scale factor and exponent.

2. What you use for a scale factor and exponent here also depends on if you are going to make an adjustment for the GBP in the RC solutions. The Intersil active filter tool does this automatically (and it does report a min GBP required which is very relatively low) while the new ADI tool offers it as an option. Doing so allows slower/ lower power/price parts to be applied.

3. The reference to that 16.8.4 section in “Op Amps for Everyone” is a bit perplexing – there is a real point that the LG collapses near Fo due to “Noise Gain” peaking and will impair the amplifier gain and output impedance around Fo – the plot is showing the response peaking which is not correct. The SKF and MFB have noise gain peaking that is always more than the filter and actually varies by RC solutions selected. The equation shown there is also perplexing – not sure where it comes from but the rule of thumb type forms shown above I believe are more common.

More of this might come out in a future article I am contemplating benchmarking tools for dymanic range. Where possible, I will also summarize reported min GBP coming out of the different tools.

5. UdyRegan
October 5, 2018

I see that this setup is far more complex than I would have otherwise expected it to be. Without prior knowledge or at least some experience in this field, things could blow out of proportion real quick. It does seem that this circuit layout would work well if the intended results need to be of a huge scale but several tweaks wouldn't hurt to suit your individual specs. I guess it does seem that managing and implementing the correct filter do indeed require some time to fix to avoid confusion along the way.

6. Tucson_Mike
October 17, 2018

He Udy, I saw your reply – not sure what specifically to say in reply to you, but it did get me reviewing my work on this over the years and comparing to the FilterPro manual (SBFA001c, 2011 I think is the latest) and what Collin references in Ch. 16 from Tom Kugelstadt. The filterpro discussion is terse, and just says more margin will be required for higher Q MFB. The full flow, and rationale, are beyond the scope of a simple comment area. However, I did uncover a simple error I had been making some years back that might be interesting – and I am pretty sure this error is imbedded in several vendor flows. The idea Collin notes is correct, the GBP margin stuff is aimed at holding a min Loop Gain around Fo typically >20dB. Lots actually goes into that (including the NG peaking – which is always higher than the filter peaking) but at very minimum, for an MFB the “Gain” that always appears in these margin equations is not the signal gain for an MFB, but 1+the signal gain. This is an inverting stage, so, if the filter gain is 1, the DC noise gain 2 for instance. Small oversight, and easy to fix.

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