Fun with third-order active filters

Butterworth response circuits come in both low- and high-pass versions, as well as ideal, Sallen-Key, emitter follower and DC accurate variants.

Many decades ago, I published a Design Idea about using equal value resistors and capacitors to implement a third-order active filter with the classic Butterworth response. This topology required two or three unity gain op-amps, depending on where the first-order RC section was located. And by swapping the positions of the resistors and capacitors, you could switch from a low-pass to a high-pass Butterworth response, while the 3 dB corner response remains the same at 1/(2*pi*R*C).

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The normalized Butterworth polynomial of S^3 + 2*S^2  + 2*S +1 can be factored into (S + 1)*( S^2 + S + 1), thereby revealing a multiplied first-order and second-order quadratic. In the original filter, the first-order section could be placed either ahead of or behind the second-order section, with the later yielding a two op-amp version; the op-amps are unity gain and provide isolation between the sections. These are the Type I versions.

A while back, I began working with different active filter topologies, including a revisit of the original Sallen-Key Type I version with an emitter follower implementation used much earlier in a car radar application. In the process, I discovered another filter version, where the factored Butterworth polynomial is directly implemented with a single op-amp. This approach, which also works with equal-value resistors and capacitors as well as an op-amp configured with a gain of 2, is called a Sallen-Key Type II. This particular configuration has a pass band gain of 2, while the Type I has a gain of 1.

The emitter follower version was implemented by replacing the unity gain op-amps in the Sallen-Key Type I with an emitter follower, and can use either two or three emitter followers similar to the op-amp version Type I. With both versions, the emitter followers can be complementary (NPN and PNP, or visa versa) which achieves a better effective input-to-output DC offset voltage, since the V_BEs cancel. Since the emitter follower has a voltage gain of slightly less than unity, this characteristic causes the amplitude response to fall between Butterworth and Bessel regions, although the phase response follows the classic Butterworth.

While continuing my investigation of various active filters, another topology popped up that apparently dates way back to early 70s Fluke DMMs (digital multimeters). Known as the DC accurate second-order low-pass filter, it was utilized as a voltage reference noise filter. This filter is quite interesting in that the filtering is achieved by shunt capacitors working against an input series resistance; the active op-amp has no resistive connection to the input or output and therefore contributes no offset voltage or bias current.

It’s also interesting (to me, at least!) that if the op-amp 2*R resistor feedback resistance is implemented with two separate series resistors of value R, along with a shunt capacitor to ground installed between them, this configuration transforms into a third-order Butterworth low-pass filter with equal-value resistors and capacitors. If the shunt capacitor to ground is then removed, the filter reverts back to a second-order Butterworth, albeit with a lower corner frequency by a factor of 2/pi. I found it quite amazing that removing a shunt capacitor to ground actually lowers the corner frequency of a low-pass filter!

Figure 1 shows the various forms of these low-pass filters for simulations, including an ideal filter version with the Butterworth transfer function.

Figure 1 The various third-order active Butterworth response low-pass filters discussed in this Design Idea include ideal, Sallen-Key Type I and II, emitter follower and DC accurate variants.

Figure 2 shows LTspice AC simulation results, illustrating the limitations of the op-amp output impedance on the stop-band rejection. Note that the emitter follower version has slightly less than unity gain and a slight deviation from the ideal response, as expected.

Figure 2 In these LTspice linear AC low-pass filter simulations, note the stop band attenuation limits due to op-amp model finite output impedance.

Figures 3a and 3b  show actual lab measurements performed with a DSO (digital storage oscilloscope)/AWG (arbitrary waveform generator) combination, utilizing the built-in Bode feature, for various low-pass filters. Compare them with the previous simulations shown in Figure 2, and note the stop band limitations due to the finite op-amp output impedance.

Figure 3 Low-pass filter Bode measurements in the lab show stop band effects due to physical op-amp output impedance limitations (a, left). In the emitter follower version’s Bode plot, note that the DC gain is -0.6 dBV (b, right).

So far so good; this is getting increasingly fun for me as I move through these various active filter topologies, and hopefully you agree! The detailed analysis for each of these filter topologies, left as an exercise for the reader, is an interesting adventure that helps illustrate what’s going on. For now, there’s more exploration to come!

Now, lets swap the resistors and capacitors in each filter topology, thereby transforming each filter from a low-pass to a high-pass version (Figure 4). The only filter to complete this transformation with any negative effects whatsoever is the DC accurate version. With this particular filter, the DC isolation due to the shunt capacitors is now replaced with shunt resistors, which obviously couple the op-amp input and output offset to the filter output. Otherwise, this filter, like its peers, behaves as expected in its high-pass form.

Figure 4 Swapping resistors and capacitors results in high-pass versions of the circuits previously seen in Figure 1.

Figure 5 illustrates the simulation results with the resistor and capacitor swaps made to each previous filter type, thereby transforming it from a low-pass to a high-pass filter with the same characteristic (i.e., Butterworth) and 3dB corner of 1/(2*pi*R*C).

Figure 5 Shown here are simulations of the high-pass filter derivations of Figure 1’s circuits, i.e., the circuits shown in Figure 4, in each case achieved by swapping resistors (R) and capacitors (C).

Figure 6a and 6b are actual lab measurements which reveal some of the measurement setup and equipment limitations in dynamic range at the low-frequency end.

Figure 6 In these lab-based Bode high-pass filter measurements (again, a at left, b at right), note the dynamic range limitations at lower frequencies.

I hope that the Bode Plot lab measurements on actual hardware for both the various low-pass and high-pass filters, for comparisons with the simulation results, are helpful for you. The circuit were built using 1% tolerance resistors and 10% tolerance film capacitors, on plug-in protoboards. You can judge for yourselves as to whether the lab measurements and simulations are in reasonable agreement; reader thoughts on this topic or anything else regarding this Design Idea are welcomed in the comments. More generally, have fun with these active filters!

Michael A Wyatt is a life member with the IEEE and has continued to enjoy electronics ever since his childhood. Mike has a long career spanning Honeywell, Northrop Grumman, Insyte/ITT/Ex-elis/Harris, ViaSat and retiring (semi) with Wyatt Labs. During his career he accumulated 32 US Patents and in the past published a few EDN articles including Best Idea of the Year in 1989.

Related Content

• Dual amplifier active filters
• Toward better behaved Sallen-Key low pass filters
• Component tolerance sensitivities of single op-amp filter sections

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