(*Editor's note*: go to http://i.cmpnet.com/planetanalog/2010/02/SCBlist.pdf for a complete, linked list of all previous installments of the *Signal Chain Basics* series.)

Analog signals are frequently filtered to band-limit noise or prevent aliasing in data acquisition systems, or to select a certain range of frequencies out of a wideband signal so that only those frequencies of interest can be operated upon. At frequencies above a few MHz, these filters usually consist of passive components, such as inductors (L), resistors (R), and capacitors (C); these are often referred to as LCR filters.

At lower frequencies, larger inductor values require physically large and often expensive inductors. This is where active filters, which combine an operational amplifier (op amp) with some resistors and capacitors, become attractive. Active filters can provide an LCR-like performance at low frequencies, and much has been written on the multitude of design options possible with these filters. Since this is a “basics” article, we’ll examine the basic concepts here to get you started with an active filter design.

Any n-th order filter may be realized by cascading the required number of first- and second-order filter stages. The transfer function of a single low-pass stage is:

The a_{i} and b_{i} coefficients determine the placement of poles and zeros for that stage. For a first-order stage, the b coefficient is always zero.

Each active filter stage is then comprised of one or more op amps and associated resistors (R) and capacitors (C). Without the use of op amps and feedback, imaginary poles could only be developed with the use of inductors and capacitors. The active filter stage gyrates the equivalent function of the inductors in the circuits, replacing them.

The two most commonly applied active filter topologies (**Figure 1**) are the Sallen-Key, which is a non-inverting, voltage-controlled, voltage-source (VCVS) topology; and the infinite-gain, multiple-feedback topology (MFB). The latter is an inverting topology in which the output polarity is inverted relative to the input.

* Figure 1. The Sallen-Key and MFB active filter topologies.
*

*(Click on image to enlarge)*

Determining the coefficients of each stage’s transfer function, and the component values needed, can be a tedious exercise of looking up coefficients in tables and performing several mathematical transformations to get to component values. Fortunately, many design tools exist to assist the designer. Texas Instruments, for example, offers the free FilterPro software to assist designing low pass, high pass, and bandpass filters, as well as some other specialized filter types.

Variations in component values from the ideal values calculated will limit the accuracy of the filter response. Because the requirements for accuracy track the filter order, these errors appear more noticeable for higher order filters. For example, when resistor values are allowed to move from the ideal to the closest 1% value, the overall gain of a sixth-order bandpass filter can drop by nearly 6 dB!

Keep in mind that capacitor values are seldom right on value, either. Therefore, for high-order filters such as sixth-order and up, plan on using 0.1% resistors and the lowest tolerance capacitors that costs and circuit performance dictate.

Another source of error that is often overlooked is selecting op amps having too low gain-bandwidth (GBW) to support the filter function. Limited gain-bandwidth may affect the filter’s cutoff frequency, gain, response curve shape, phase and transient behavior. A good rule of thumb is to select an op amp with a GBW at least one hundred times the product of the stage’s natural frequency (f_{n}) and Q. Some tools, such as FilterPro, will recommend the GBW needed for the op amps used in each stage.

Join us next month when we will talk about methods for saving power in home audio applications.

**References**

· Kugelstadt, T., “Active Filter Design Techniques”, Chapter 16 of *Op Amps for Everyone*, 2009, Elsevier Inc.

· Huelsman, L.P., Allen, P.E., *Introduction to the Theory and Design of Active Filters*, 1980, McGraw-Hill.

**About the author**

*Rick Downs* is signal chain applications manager for Texas Instruments’ Analog eLab, which provides analog design tools online. Over the past 25 years, Rick has held various positions in applications and marketing of analog semiconductors focused on audio, data acquisition, digital temperature sensors and battery management products. Rick received his BSEE from the University of Arizona, and holds four patents. He has authored several articles and application notes on analog topics, and prepared and delivered several seminars on data acquisition. You can send your questions to Rick at scb@list.ti.com.