Technical Article

Inductor Out, Op-Amp In: An Introduction to Second-Order Active Filters

May 19, 2016 by Robert Keim

In this article, we’ll compare active and passive filters and look at some common second-order active topologies.

In this article, we’ll compare active and passive filters and look at some common second-order active topologies.

Supporting Information

Active vs. Passive

If your filter consists of nothing more than resistors, capacitors, and inductors, you have a passive filter. The circuit becomes “active” when you incorporate an active component, e.g., a transistor. Theoretically, you could design an active-filter circuit using an individual transistor in conjunction with passive components, but in practice, the active component of choice is an operational amplifier. Op-amps offer performance advantages over individual transistors, and they also simplify the process of designing and analyzing a filter circuit. So as you read this article, keep in mind that for all practical purposes “active filter” means “op-amp-based active filter.”

Passive ≠ Bad

It is important to understand that active filters are not inherently “better” than passive filters. On the contrary, I prefer passive filters and use them whenever possible. Some advantages of the old-fashioned approach are the following:

  • There is no need to worry about the nonideal characteristics of the op-amp—offset voltage, bandwidth limitations, noise. . . .
  • Breadboarding or PCB layout is simpler and cleaner without the power and ground connections required by the op-amp.
  • Passive circuits are more straightforward and hence less subject to design errors—for example, compare a resistor-inductor-capacitor (RLC) low-pass filter (see the next section) to an equivalent Sallen–Key circuit (scroll down to the “­­­­­­­­­­­­Sallen–Key” section).

Active filters certainly have their advantages though. The most prominent benefit that applies to both first-order and second-order filters is the improved impedance characteristics. Op-amps provide high input impedance and low output impedance, and thus an op-amp-based active filter can outperform a passive implementation when the incoming signal has relatively high source impedance or when the output signal must drive relatively low load impedance.

Another advantage is gain: If the signal needs to be not only filtered but also amplified, you really have no choice but to use an active filter—either a specific active-filter topology or a passive-filter-plus-amplifier arrangement.

Before we continue, I should point out that it is certainly possible to create a second-order active filter that consists of an op-amp and two first-order filters. The two filter stages are connected in series, with the op-amp buffering the output of the first stage. These “cascaded” filters inevitably produce a gradual transition from passband to stopband, resulting in nonlinear phase response and significant attenuation of signals near the end of the passband. The two second-order topologies discussed below are usually preferable because they allow you to optimize an individual circuit for sharper transition from passband to stopband, minimal passband attenuation, or linear phase response.

The Nefarious Inductor

As indicated by its title, this article focuses on second-order active filters, i.e., filters that have two poles in their transfer functions and thus achieve steeper roll-off. Passive filters need two energy storage elements—a capacitor and an inductor—to provide a second-order response . . . and this is where the trouble begins. Here is a second-order RLC low-pass filter, with equations for the cutoff frequency (fc) and the quality factor (Q):



\[f_c=\frac{1}{2\pi\sqrt{LC}}\ \ \ \ \ \ \ \ \ Q=\left(2\pi f_c\right)\times CR\]


This otherwise respectable filter is tainted by its association with the inductor. The fact is, inductors are downright unpopular, and here’s why:

  • They’re bulky, and as you have probably noticed, electronics manufacturers want to make their widgets smaller, not bigger.
  • Inductors are not particularly compatible with integrated-circuit manufacturing techniques:
    • You can’t get much inductance out of an IC inductor, which means the cutoff frequency of the filter can’t go very low.
    • IC inductors are seriously nonideal; the various parasitic impedances of the IC environment are more problematic than those experienced by discrete inductors.
  • Inductors generate more electromagnetic interference (EMI) than resistors and capacitors, and they are also more susceptible to EMI.

The clear conflict between inductors and the trends that dominate the electronics industry—miniaturization, monolithic fabrication, wireless functionality—is a major motivation for pursuing second-order filters that do not require inductance.

Antoniou and His Simulated Inductor

One way to avoid the problems associated with inductors is to use a circuit that behaves like an inductor yet requires only resistors, capacitors, and op-amps. The following “inductance-simulation circuit” was invented by Andreas Antoniou:



\[equivalent\ inductance:\ L=\frac{R_1R_3R_4C_1}{R_2}\]


How Professor Antoniou ever figured this out is beyond me. In any event, I’m not going to dwell on this circuit because the Sallen–Key and Multiple Feedback (MFB) topologies are a simpler and more direct route to second-order filter performance. It’s good to be aware, though, that various RLC filters can be implemented without inductors by using an inductance-simulation circuit.


A Sallen–Key filter gives you two poles with only one op-amp and a few passives. The following is a Sallen­–Key implementation of a unity-gain low-pass filter.





It is often the case that there is no need to amplify any portion of the input signal; the filter is there to suppress unwanted frequencies, and it’s fine if the frequencies of interest merely pass through. These unity-gain applications are common enough to make the Sallen–Key a very popular filter, despite the fact that the MFB topology is advantageous when the gain becomes significantly higher than unity.

Let’s think about what happens at low frequencies. C1 and C2 become open circuits, and the resistors become irrelevant because the current flowing into the op-amp’s positive input terminal is negligible. Thus, we are left with a voltage follower. This means that 1) the Sallen–Key filter does not invert the signal and 2) the gain will be almost exactly unity without any dependence on component values. As you will see in the next section, the gain of the MFB circuit is determined by component values, even at unity gain, and this explains why the Sallen–Key topology is preferred for unity-gain applications.

For much more information on the Sallen–Key topology, click here (PDF) for a Texas Instruments app note that just might tell you everything you ever wanted to know about op-amp-based active filters. Another valuable resource is this online filter design tool, which will help you with Sallen–Key low-pass and high-pass circuits.

Multiple Feedback

Here is an MFB low-pass circuit:





\[DC\ gain\ =\ \frac{R_3}{R_1}\]


If you replace the capacitors with open circuits and ignore R2 (again, because the input current is negligible), you will recognize the standard op-amp inverting configuration:



Thus, MFB is an inverting topology. You might recall that there is no inverting version of a voltage follower; if you need a unity-gain inverting op-amp circuit, you have to use an inverting amplifier with R1 = R3. The same applies to the MFB topology: for unity gain, you set R1 = R3, which means that the accuracy of your gain depends on the precision of your resistors. As the gain increases, though, an MFB circuit actually becomes less sensitive to component tolerances than an equivalent Sallen–Key implementation, so MFB is usually a better choice for higher-gain filters. The app note mentioned in the previous section is also a great resource for MFB circuits, and the same online filter tool can help you with MFB filter design.


We’ve covered quite a bit of introductory information related to why we use second-order active filters and how we create second-order circuits using a single op-amp in conjunction with capacitors and resistors. However, we’ve only scratched the surface of this expansive subject. Keep an eye out for future articles that explore these and related topics in greater detail.