Filters are circuits comprising resistors, inductors and capacitors that pass certain frequency bands and attenuate others. Active filters use operational amplifiers (op amps) with resistors and capacitors, whereas passive filters do not include an amplifier.
Active filters have an advantage over passive filters in that they can have gain. The gain factor in an active filter enables higher-order filters as well as a wider range of filter-response characteristics. For example, a band-pass filter with a narrow bandwidth would be challenging to implement with only passive components given the limited gain of passive filters.
The amplifier also allows the filter to drive a load impedance without affecting the filter response, whereas the passive filter response generally changes significantly with loading. Finally, a passive filter may require an inductor, which may add cost and PCB space, compared to an active filter that only requires the amplifier, resistors, and capacitors.
Generally, you can design active filters with simple software programs such as Filter Design Tool. This tool designs an active filter based on your filter specifications, type, response, and topology. The final output for the software tool is a schematic with all the necessary component values. Filter Design Tool does assume that the op amp is ideal, however, and its response can change when using a practical op amp.
This article covers the most common active filter nonidealities, with recommendations on how to mitigate the problem.
Gain-bandwidth limitations
As mentioned previously, the purpose of an active filter is to pass one range of frequencies (pass band) while attenuating another range of frequencies (stop band). The ability of an amplifier to achieve this goal depends heavily on the gain-bandwidth product specification. When using an op amp with insufficient gain bandwidth in an active filter, the stop-band attenuation will not match the expected ideal filter response; specifically, the gain will stop rolling off and will begin to increase rather than decrease.
Figure 1 shows the simulated gain response of a Sallen-Key active filter using three different op-amp models with gain bandwidths ranging from 1.2 MHz to 11 MHz. The term Sallen-Key describes the topology or the specific connections of circuit components. Here, Filter Design Tool recommends a minimum gain bandwidth of 7.1 MHz, so op amps like OPA170 and OPA340 don’t meet that requirement but OPA140 does.
Figure 1 Sallen-Key low-pass filter response is shown for different gain-bandwidth op amps. Source: Texas Instruments
The attenuation in the stop band should continue to roll off at –40dB/decade; however, once the amplifier reaches its gain-bandwidth limit, it stops attenuating. The larger the gain-bandwidth of the op amp, the better the stop-band attenuation.
The figure above showed the effects of insufficient gain bandwidth on a Sallen-Key topology. The Sallen-Key topology is a noninverting type of active filter, while the multiple-feedback topology is an inverting type of active filter. The multiple-feedback topology band-stop attenuation is better than the Sallen-Key topology for low-gain-bandwidth designs.
Figure 2 shows the multiple-feedback response for the same three op amps used in the Sallen-Key example. The multiple-feedback response continues to roll off even after the frequency exceeds the gain-bandwidth limitation of the op amp. For the multiple-feedback circuit, the rolloff transitions from 40dB/decade to 20dB/decade when reaching the bandwidth limitation.
Figure 2 Multiple-feedback low-pass filter response is shown for different gain-bandwidth op amps. Source: Texas Instruments
To summarize, it’s important for active filters to follow the gain-bandwidth limitation given by the filter design software. A higher gain bandwidth will improve the stop-band attenuation, so depending on the range of frequencies that you need to reject, it may be necessary to choose a gain bandwidth 10 or 100 times the recommended minimum.
In cases where it’s not practical to choose an amplifier with a high gain bandwidth, use the multiple-feedback topology because its response is less dependent on gain bandwidth than the Sallen-Key topology.
Source impedance effects
All signal sources have a finite output impedance. A signal source can be commercial test equipment, an amplifier output, or an analog sensor. The output impedance of test equipment or amplifiers is generally very low, while the output impedance of a sensor can be high. Sources with low output impedance will have a minimal effect on filter response.
Conversely, a signal source such as a bridge sensor can have a high output impedance, which may significantly affect the filter response. As with the gain-bandwidth limitations, some filter topologies will be more sensitive to source impedance than others.
Figure 3 compares and contrasts the impact of source impedance on the Sallen-Key and multiple-feedback topologies. In this case, Sallen-Key is much less sensitive to source impedance than multiple feedback. The example plots the response for a 250 Ω, 500 Ω and 1,000 Ω source impedance. The multiple-feedback topology gain is significantly affected by the source impedance, but the effect on the Sallen-Key topology is minimal.
Figure 3 Source-impedance effect on response is displayed for the Sallen-Key and multiple-feedback topologies. Source: Texas Instruments
Component tolerance
Resistor tolerance will affect the cutoff frequencies and gain of active filters. For a low- or high-pass filter, a small shift in the cutoff frequency will not be a significant problem for most applications. For example, if a low-pass filter has an ideal cutoff frequency of 1 kHz but the component tolerance sets the actual frequency to 1.1 kHz, that will not be an issue for most applications.
Band-pass and band-stop filters, conversely, often have very precise bandwidth and center frequency requirements. For example, if a band-stop filter is designed to attenuate a 60-Hz signal, then the center frequency of the stop band should be very close to 60 Hz in order to properly attenuate the 60-Hz signal.
Figure 4 illustrates how component tolerance can affect the center frequency of a band-pass filter. The filter has a center frequency of 1 kHz and a bandwidth of 100 Hz. The circuit was tested using Monte Carlo analysis with good and poor component tolerance (good tolerance: RTOL = 0.1%, CTOL = 1%; poor tolerance: RTOL = 1%, CTOL = 5%). Monte Carlo analysis randomizes the component values according to the assigned tolerance for a set number of runs.
Figure 4 Here is a view of component tolerance impact on band-pass center frequency. Source: Texas Instruments
In this example, you can see much better control of the band-pass center frequency for the more precise (good) tolerance case. For the poor tolerance case, the center frequency is between 932 Hz and 1.106 kHz. The goal of this filter is to pass 1 kHz and reject all other frequencies. For many of the poor-tolerance Monte Carlo runs, a 1-kHz signal would be significantly attenuated because of the shifted center frequency.
The problem with component tolerance is most significant on band-pass and band-stop filters. There are two ways to avoid tolerance issues for these filters. The obvious approach to minimize the error is to choose resistors and capacitors with good tolerance. Generally, precision resistors of 0.1% are easy to find and not that expensive compared to the amplifier cost.
For capacitors, the C0G or film type have the best tolerance, temperature coefficient, and distortion. Unfortunately, these capacitors can be expensive, and the best tolerance is typically 1%. Furthermore, the maximum capacitance for C0G capacitors is generally 100 nF.
Another approach to minimizing the effect of component tolerance is to widen the bandwidth of the filter. The example shown in Figure 4 uses a 100-Hz bandwidth. Running the same example with a 400-Hz bandwidth would show that both the poor and good tolerance cases would pass the 1-kHz center frequency. However, the disadvantage of increasing the bandwidth is that the stop-band attenuation would not be as effective for the wide-bandwidth case.
Noise gain in active filters
All amplifiers and resistors generate noise because of the random motion of electrons inside the resistors and the semiconductor material. It’s possible to calculate this noise mathematically based on the circuit components.
The noise for an op amp is defined as a noise signal source on the noninverting input of the op amp. The gain seen by the noise signal source is called the noise gain. For active filters, the noise gain and noise bandwidth can be significantly different than the signal gain and bandwidth of the filter.
Figure 5 illustrates a multiple-feedback low-pass filter signal gain and noise-gain response. The top schematic shows the signal gain of the filter, while the bottom schematic shows the noise-gain configuration. To simulate noise gain, you simply add a signal source to the noninverting input and short out other voltage sources.
Figure 5 See the noise gain vs. signal gain for a multiple-feedback low-pass filter. Source: Texas Instruments
Comparing the noise gain and signal gain on the Bode plot shows that the noise gain has a much wider and higher frequency response than the signal gain. Sometimes engineers conducting noise analysis of filters assume that the filter transfer function will limit the noise and are surprised that the noise is higher than expected.
This unexpected increase in noise is because the op amp’s intrinsic noise source is located at the noninverting input and not at the signal input. Thus, the increase in noise gain and noise bandwidth causes an increase in output noise that engineers often overlook because they are considering signal gain and not the noise gain.
As the above figure shows, a low-pass active filter has a higher and wider frequency noise gain than the signal gain. This effect is even more significant on band-pass and band-stop filters. Figure 6 compares the noise gain and signal gain of a band-pass filter with a 100-Hz bandwidth. In this case the noise gain is about 46dB (200 V/V) higher than the signal gain.
Figure 6 Noise gain vs. signal gain is shown for a band-pass filter. Source: Texas Instruments
This high noise gain will significantly boost the op-amp noise. But engineers who analyze the circuit by just looking at the signal gain may not expect the higher output noise because of the noise gain.
One way to reduce the impact of noise gain for band-pass and band-stop filters is to increase the filter bandwidth. This is counterintuitive because noise is usually larger for wide-bandwidth circuits. However, for bandpass filters, the noise gain is higher for narrow-bandwidth filters and lower for wide-bandwidth filters. Figure 7 shows how noise gain is higher for the narrow-bandwidth filter.
Figure 7 Noise gain vs. noise bandwidth is shown for band-pass filters. Source: Texas Instruments
Resources for active filters
Fully understanding active filters is a deep and relatively complex subject. You can use Filter Design Tool to design active filters, because this tool and other filter design tools generally assume ideal op-amp characteristics, but the best practice is to confirm your design using a SPICE simulator.
Art Kay is an application engineer in the precision amplifiers team at Texas Instruments. He specializes in support of low noise data acquisition systems and has published a book on intrinsic noise analysis. He also co-published a useful summary of analog engineering relationships and a companion software tool.
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