Active filters are a common and extremely useful building block for electronic circuits. They come in so many varieties, with so many descriptors that it can become confusing. Some terms are easy to grasp – low-pass, band-pass or high-pass for example are self-evident. It’s the same with different circuit topologies such as Sallen & Key, VCVS, switched capacitor. But what about the terms “Butterworth”, “Chebyshev” or “Bessel” when applied to filters? Do you know what they mean?
These terms describe the characteristics of a filter independent of the circuit topology, the number of poles or filter type. I will be using the example of a 4-pole VCVS low-pass filter as shown in Figure 1. This filter is made up of two identical two-pole stages in series. Let’s assume we want a corner frequency, fc , of 10kHz, but what other specifications do we need to consider?
This is a 4-pole VCVS low-pass filter built from two identical 2-pole sections in series. A high-pass version of this filter is similar, except the capacitors C1x and resistors R1x are swapped. You can cascade any number of such stages to increase the number of poles.
We can think about a filter’s performance in the frequency-domain (gain and phase shift vs frequency) and in the time-domain (how the filter output voltage responds to a step input vs time). Figure 2a shows the generic frequency-domain characteristic of a 4-pole filter. Below the corner frequency fc , is the passband where we expect the filter attenuation to be zero. The filter may introduce some ripple (shown exaggerated in the diagram) in the pass band which may be important depending on our application. Above fc is the stopband where the gain of the filter falls off at 80dB per decade (since we have a 4-pole filter). Between these is the transition region. Our choice of component values will impact all of these.
The generic frequency-domain gain of a 4-pole low-pass filter is shown on the left. In a practical implementation we can control the ripple in the passband and the characteristic of the transition region by careful component selection. The time-domain step-response of the filter is shown at right. The rise time, overshoot and settling time can similarly be controlled by design.
Figure 2b shows the time-domain step-response of the filter. Because the filter introduces a frequency-dependent phase shift, the step-response will have some finite rise-time tr. There may also be some overshoot (usually expressed in percent) and maybe some ringing which settles over some settling time ts . Again, depending on your application, these characteristics may or may not be important, but are influenced by component choice.
You have probably guessed by now, but the filter realisations described above optimise for one or more of these characteristics; A Butterworth filter has a maximally flat passband, at the expense of some overshoot and a fairly gentle knee at the corner frequency. A Chebyshev filter on the other hand, has a sharper transition from passband to stopband at the cost of some ripple in the passband and a good deal of overshoot and ringing in the time-domain. A Bessel filter is optimised for step-response at the expense of a very gentle transition at the corner frequency.
The mathematics behind this complex, but fortunately we don’t have to bother with it. Traditionally you would calculate the component values from a table in a filter design handbook, but more likely today you would use an on-line filter design tool. I’m going to do it the old-school way using a table and simulate the resulting filters in LT-Spice. The design table I will use comes from Horowitz and Hill’s “The Art of Electronics”. I have reproduced a small part of it relevant to 4-pole filters in Table 1 below. These tables normally provide different options for the Chebyshev filter allowing you to choose the maximum passband ripple you can tolerate. I chose 2dB for this example.
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Each row of the table gives a time-constant scale factor Cn and a stage gain K for each type of filter. Let’s start with the Butterworth. First, we calculate the time constant of each stage of the filter from Equation 1 (in the case of the Butterworth they will be identical).
We get 15.916 µs with our example. Choosing C1x to be 1 nF, R1x must be 15.916 kΩ. Now we have to select the gains for each stage. We choose Rx to be 10 kΩ so, using the values from the table RGA must be 1.520 kΩ and RGB must be 12.350 kΩ. The overall gain of the circuit at DC will be given by the combined gain of the two stages. Repeating this for the Chebyshev, and Bessel filters gives the values in Table 2 below.
Plugging these values into LT-Spice gives the normalised frequency response curves shown in Figure 3. The solid line is the gain, and the dotted line is the phase shift. The Butterworth curve (red trace) has a flat passband as we would expect and a classic roll-off reaching -80dB in exactly one decade at 100kHz. The Chebyshev curve (yellow trace) has a much steeper transition but the expected passband ripple. The Bessel filter (green trace) on the other hand has a gentler roll-off than either of the others.
These frequency-domain plots show the significant differences between filter realisations. Compared to the Butterworth realisation (red trace) the Chebyshev realisation has a much sharper transition, but this comes with the increased passband ripple. The Bessel realisation has an even softer transition because it is optimised for time-domain response.
The normalised step responses shown in Figure 4 gives a different perspective. The Bessel filter (red trace) has a fast rise time, little overshoot and fast settling time. The Chebyshev filter (blue trace) has the worst step response performance with a slower rise time, around 25% overshoot and a lot of ringing. Table 3 below summarises the measured performance of my simulation.
These time-domain responses show how the Bessel realisation is optimised for rise time, overshoot and settling time. In contrast, the Chebyshev response, which has the sharpest frequency-domain transition, has the slowest rise time, about 25% overshoot and the longest settling time.
You can see that the filter realisation has a big impact on filter performance, even with the same circuit topology. Choosing the right one for your application is important. Armed with this knowledge and a good filter design tool (or handbook) you should be able to optimise your filter realisation to suit your application.
References
“LTspice Information Center | Analog Devices.” Accessed February 25, 2023. https://www.analog.com/en/design-center/design-tools-and-calculators/ltspice-simulator.html.
Horowitz, Paul, and Winfield Hill. The Art of Electronics. Third edition, 11th printing, with Corrections. Cambridge New York, NY: Cambridge University Press, 2017.
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Andrew Levido (andrew.levido@gmail.com) earned a bachelor’s degree in Electrical Engineering in Sydney, Australia, in 1986. He worked for several years in R&D for power electronics and telecommunication companies before moving into management roles. Andrew has maintained a hands-on interest in electronics, particularly embedded systems, power electronics, and control theory in his free time. Over the years he has written a number of articles for various electronics publications and occasionally provides consulting services as time allows.
