Butterworth Filter Includes:
Butterworth filter Butterworth calculations
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The Butterworth filter topology is widely used in many RF and general filter applications.
One of the key features of the Butterworth filter is that it has what is termed a maximally flat response within its pass-band.
In fact, the Butterworth filter is often considered as a good all round form of filter which is adequate for many applications, although it does not provide the sharpest cut-off.
Butterworth filter development
In the early days of wireless technology, filter technology was not nearly as advanced as it is today. Research into filters and how they could be realised and their performance predicted was undertaken in many areas.
In one development Stephen Butterworth of the Admiralty Research Laboratory in the undertook the development of a filter that produced a flat response within the pass-band. Butterworth published a paper on his work in the UK in October 1930. The paper was entitled: "On the Theory of Filter Amplifiers" and in it he developed the basic equations for a maximally flat filter for use within RF valve amplifiers.
The article was published in a magazine entitled Experimental Wireless and Wireless Engineer. This title was published in the UK by Iliffe and Sons in the 1920s and early 1930s, later changing its title to:" Wireless Engineer and Experimental Wireless."
In the article Butterworth stated:
Apart from the compactness of the system, the filter amplifier has an advantage over orthodox systems in that the effect of the resistance is under complete control so that we may construct filters in which the sensitivity is uniform in the pass region.
At the time, much of this technology was relatively new, and in addition to this, it would be many years before computer technology was available to analyse circuits. As a result, filter designs tended to exhibit large levels of in-band ripple and this was a problem when people needed flatter responses. Butterworth was the first person to be able to achieve a nearly flat in-band response.
In his paper, Butterworth produced equations for two- and four-pole filters. However to minimise the actual loss of the filter, he showed how further sections could interspersed with thermionic valve, vacuum tube amplifiers.
Butterworth filter amplitude response
As mentioned above, the key feature of the Butterworth filter is that it has a maximally flat response within the pass-band, i.e. it has no response ripples as in the case of many other forms of RF filter.
There is a frequency known as the cut-off frequency. This is defined as the point on the Butterworth filter response where the power drops to half, i.e. the voltage drops to 71%, i.e. 1/√2 of its maximum amplitude at lower frequencies. It is also worth noting that the maximum amplitude , i.e. minimum loss for the Butterworth filter response occurs at 0 Hz or radians/s.
When plotted on logarithmic scales, the Butterworth filter response is flat within its pass-band and then rolls off with an ultimate linear roll off rate of -6 dB per octave (-20 dB per decade). A second-order filter decreases at -12 dB per octave, etc. The ultimate roll off rate is actually the same for all low pass and high pass filters of the same order regardless of the filter type.
When compared to other forms of filter such as the Chebyshev or elliptic filter formats, the Butterworth filter reaches its ultimate roll-off rate more slowly. In fact the Butterworth filter was derived on the basis that the behaviour below the cut-off frequency was more important than at any other frequency. This means that it is good for audio applications. However it means that it has a tolerably good amplitude response and good phase response, although the performance around the cut-off frequency is poor.
Butterworth filter phase response
A further advantage of the Butterworth filter is that Butterworth filters have a more linear phase response in the pass-band than types such as the Chebyshev or elliptic filters, i.e. the Butterworth filter is able to provide better group delay performance, and also a lower level of overshoot .
Butterworth filter impulse response
The Butterworth filter may also be judged in terms of its time domain response including its response to impulses. It has a response that gives an increasing level of overshoot with increasing filter order. For a fourth order filter, i.e. n = 4, the level of overshoot exceeds 11%.
When analysing the optimum filter format, it is best to analyse the different advantages and disadvantages of the different types. The Butterworth provide a flat un band response and a more linear phase response than many others.
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