Butterworth RF Filter Basics

- basics of the Butterworth Filter - its performance, key facts and how it can be used in RF filter applications.

Butterworth RF filter tutorial includes:

    •  Butterworth filter
    •  Formulae and Calculations

The Butterworth filter is a form of RF filter using lumped elements that is widely used in many radio frequency filter applications.

The key feature of the Butterworth filter when compared to other forms of filter is that it has a nominally flat response within its pass-band and an adequate roll-off.

As a result the Butterworth filter may also be known as the maximally flat magnitude filter.

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 emergence

The Butterworth filter gains its name from the first person to develop the theory for the filter. Stephen Butterworth of the Admiralty Research Laboratory in the UK published his paper in October 1930, entitled: "On the Theory of Filter Amplifiers" in which 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, filter designs exhibited large levels of in-band ripple and this was a problem when people needed flatter responses. Butterworth was accordingly 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.

Butterworth Filter Cut-off

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. 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 only 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 Phase Response

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%.

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