The Butterworth filter can be realized using passive components such as series inductors and shunt capacitors with Cauer topology – Cauer 1-form as shown in the figure below. But, Cauer topology is typically used for passive realization and Sallen-Key topology is typically used for active realization.īutterworth Filter Design using Cauer Topology In general, there are various topologies used for implementing the linear analog filters. The above equation can be represented in S-domain as given below Where, Vout indicates voltage of output signal, Vin indicates input voltage signal, j is square root of -1, and ‘ω’ = 2πƒ is the radian frequency. Now, if we transpose the above equation, then we will getīy using the standard voltage transfer function, we can define the frequency response of Butterworth filter as Where, H0 represents the maximum pass band gain and H1 represents the minimum pass band gain. But, if we want to define Amax at another voltage gain value, consider 1dB, or 1.1220 (1dB = 20logAmax) then the value of ε can be found by: If we define Amax at cut-off frequency -3dB corner point (ƒc), then ε will be equal to one and thus ε2 will also be equal to one. Where ‘n’ indicates the filter order, ‘ω’ = 2πƒ, Epsilon ε is maximum pass band gain, (Amax). The quality factor for this type is just Q=0.707 and thus, all high frequencies above the cut-off point band rolls down to zero at 20dB per decade or 6dB per octave in the stop band. The Butterworth filter has frequency response as flat as mathematically possible, hence it is also called as a maximally flat magnitude filter (from 0Hz to cut-off frequency at -3dB without any ripples). The corner frequency or cutoff frequency is given by the equation: Cutoff Frequency
The filters are used for shaping the signal’s frequency spectrum in communication systems or control systems. There are various types of Butterworth filters such as low pass Butterworth filter and digital Butterworth filter. Hence, this type of filter named as Butterworth filter. In 1930 physicist and the British engineer Stephen Butterworth described about a Butterworth filter in his “on the theory of filter amplifiers” paper for the first time. The signal processing filter which is having a flat frequency response in the passband can be termed as Butterworth filter and is also called as a maximally flat magnitude filter.