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.
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).
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.
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.
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.
In Partnership with Allegro MicroSystems
by Jeff Child
Rob said:
“…….. 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. . . . . . .”
NONSENSE! The two poles are either real or (more generally, for reasonable Qs) complex conjugates. (So – which is the low-pass pole and which is the high-pass pole!). The 20 db asymptotic roll-offs are due to the ZEROS; one at s=0 and the other at s=infinity. Your “explanation is just wrong.
- Bernie