# Op-Amps as Active Band-Pass and Active Band-Reject Filters

January 24, 2020 by Robert Keim
In this video, we will continue our discussion of op-amp-based active filters by examining band-pass and band-reject topologies.
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We learned in the previous tutorial that op-amps allow us to design high-performance filters that provide second-order frequency response without the need for inductors. This is beneficial because, in the context of modern electronic design, the characteristics of inductors are significantly more problematic than those of resistors and capacitors.

## Band-Pass and Band-Reject

A low-pass filter passes frequencies that are below the cutoff frequency, and a high-pass filter passes frequencies that are above the cutoff frequency. A band-pass filter, in contrast, passes frequencies that fall only within a relatively narrow range, and a band-reject filter (also called a band-stop or notch filter) passes all frequencies except those that fall within a relatively narrow range.

Band-pass filters are widely used in communications systems, because they can separate a received signal from other received signals that occupy adjacent frequency bands. Band-pass filters can also be used to identify the pitch of an incoming audio signal. Band-reject filters are useful when a system needs to suppress noise that consists of a small range of frequencies (such as interference from a variable-frequency oscillator) or of a single frequency (such as 60 Hz power-line interference).

## Single-Pole vs. Double-Pole Roll-Off

In the previous tutorial, we saw that the Sallen–Key filter requires only one op-amp and can generate a second-order—i.e., a two-pole—response. Another two-pole op-amp-based architecture is the multiple-feedback (MFB) topology.

When a Sallen–Key or MFB filter implements a high-pass or low-pass response, the maximum roll-off is 40 dB/decade. However, when these two-pole topologies are used for a band-pass or band-reject response, the roll-off will eventually stabilize at only 20 dB/decade, because the passive components are designed in such a way as to dedicate one pole to the low-pass portion of the frequency response and one pole to the high-pass portion of the frequency response.

In the following sections, we will look at two-pole active band-pass and band-reject filters. These circuits require only one op-amp. Keep in mind, though, that band-pass and band-reject filters with steeper roll-off can be achieved by combining a Sallen–Key for MFB low-pass filter and a Sallen–Key or MFB high-pass filter. When a band-pass response is needed, you can use a high-pass filter followed by a low-pass filter. When a band-reject response is needed, you can use a summation stage to add a low-pass-filtered signal to a high-pass-filtered signal. The cutoff frequencies of the two filters are adjusted such that the frequency responses overlap in a way that creates a passband or a notch.

## MFB Band-Pass Filter

The following schematic shows an op-amp-based active filter that creates a band-pass response.

The center frequency is calculated as follows:

$f_{CTR}=\frac {\sqrt {\frac {1}{(R_1||R_2)R_3C_1C_2}}}{2\pi}$

Though the roll-off will tend toward 20 dB/decade, the roll-off near the center frequency can be significantly steeper, because this portion of the frequency response is influenced by the filter’s Q factor. A higher Q increases the roll-off near the center frequency—in other words, a higher-Q band-pass filter is a more selective filter.

## Band-Reject Filter

A high-Q band-reject active filter is shown below. This topology is called a twin T network.

This schematic includes component values that result in a notch frequency of 60 Hz; the rejection of 60 Hz power-line interference is a standard notch-filter application. Notice that in this schematic the component values are chosen such that R1 = R2, R1 = 2×R3, C1 = C2, and C1 = C3/2. The notch frequency is calculated as follows:

$f_{NOTCH}=\frac {1}{2\pi R_1C_1}$

The configuration shown above demonstrates clearly that the circuit consists of two T-shaped passive-component networks; the network composed of two resistors and a capacitor acts like a low-pass filter, and the network composed of two capacitors and a resistor acts like a high-pass filter. Thus, the circuit has a low-pass stage and a high-pass stage working in parallel, and it’s interesting to note that the band-reject implementation described earlier in this article consists of a low-pass-filtered output and a high-pass-filtered output combined via summation.

## Summary

• Band-pass and band-reject filters are used to pass or block a narrow range of frequencies.
• It is possible to create a band-pass or band-reject response using only one op-amp in conjunction with capacitors and resistors.
• An alternative approach to band-pass and band-reject design is to combine the response of an active low-pass filter and the response of an active high-pass filter. However, this leads to increased component count: at least two op-amps are required, and the number of passive components is significantly higher.