![]() The equal angular spacing of the Butterworth poles indicates that even-order filters will have only complex-conjugate poles.In the example above, N = 4, and the separation angle is 180°/4 = 45°. The angle that separates the poles is equal to 180°/N, where N is the order of the filter.Thus, the distance between the origin and each pole is the same, and this in turn means that all poles have the same frequency. All points on a circle have the same distance from the center of the circle.The fundamental characteristic of a low-pass Butterworth pole-zero plot is that the poles have equal angular spacing and lie along a semicircular path in the left half-plane.The following bullet points will help you to unpack the information contained in this diagram. Additional poles would need to follow the same pattern. To achieve a low-pass Butterworth response, we need to create a transfer function whose poles are arranged as follows: Passband flatness is evident in the following plot, which is the magnitude response of a fourth-order Butterworth filter. ![]() This contrasts with the Chebyshev topology, which allows passband ripple in order to increase the steepness of the transition from passband to stopband. The Butterworth priority is passband flatness, and that is what unites the various instantiations of the Butterworth topology: they minimize the amount of magnitude variation that occurs prior to the cutoff frequency. ![]() Instead, we have trade-offs: performance vs. I use the word “topology” here to emphasize the fact that the Butterworth “filter” is actually a class of circuits that have the same general characteristics.Īs with most other things in life, you can’t have one system or device or material that is better than all others in every way. As you can see in the diagram below, it indicates both pole/zero frequency and Q factor: In the example above, the two poles represent a complex-conjugate pair, because they have real parts that are equal and imaginary parts that are equal in magnitude but opposite in sign.Ī pole-zero plot is a convenient and effective means of conveying important information about a filter system. Poles are marked with an ✕, and zeros are marked with a circle. The location of a pole or zero is determined by its real part, which is plotted horizontally, and its imaginary part, which is plotted vertically. The following diagram demonstrates the basic structure. Sergio Franco’s article on pole splitting, you are familiar with the concept of a pole-zero plot. If you’ve read my article on the Nyquist plot or on complex-conjugate poles, or Dr. Information about a system’s poles and zeros can be conveyed visually by marking their locations in the complex plane. (All low-pass filters have at least one zero at ω = infinity, but these don’t appear in the pole-zero diagram and can usually be ignored.) In this article, we will focus on the Butterworth low-pass filter, which has at least two poles and no zeros. Zeros represent frequencies that cause the numerator of a transfer function to equal zero, and they generate an increase in the slope of the system’s transfer function. Poles represent frequencies that cause the denominator of a transfer function to equal zero, and they generate a reduction in the slope of the system’s magnitude response. I previously wrote an article on poles and zeros in filter theory, in case you need a more extensive refresher on that topic. My objective in this article is to help you understand the Butterworth filter by presenting and discussing aspects of its pole-zero diagram. In the days of SPICE and Google searches and filter calculators, we sometimes need to take a step back and think about the theoretical and conceptual foundation upon which a functional circuit is built. However, in some cases, it is beneficial to understand things more thoroughly. You certainly don’t have to be an expert in filter theory to successfully incorporate a filter topology into your designs. Many people have heard the term “Butterworth filter” and have used these types of filters in their circuits.
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