Butterworth Filtre / Butterworth Filter

In applications that use filters to shape the frequency spectrum of a signal, such as in communication or control systems, it is also called the "transition band". The rounding shape or width can be too long or wide for a simple first-degree filter. Therefore, active filters designed with more than one "order" are required. Such filters are often known as "high-order" or "n. level" filters.

Complexity or filter type is defined by the "order" of filters. It depends on the number of reactive components in its design, such as capacitors or inductors. We also know that the rolling rate and therefore the width of the transition band depends on the degree number of the filter and has a rounding rate of 20db/decade or 6db/octave for a simple first-degree filter.

Then, for a filter with a number sequence of n., 20n dB/decade or 6n dB/octave will have a subsequent roll-off ratio. Therefore, a filter of the first order is 20db/decade (6dB/octave), a filter from the second order is 40dB/decade (12dB/octave), and a filter from the fourth order is 80db/decade (24dB/octave) etc…

High-level filters such as the third, fourth and fifth order are usually created by combining filters from the first order and the second order.

For example, two brothers can be cascaded together to produce a low-grade pass filter, a lower-than-fourth-degree transition filter, and so on. Although there is no limit to the order of the filter that can be created, the size and cost increase as the order increases. In proportion to this, its accuracy decreases.

Logarithmic Frequency Scale

Butterworth Filter
Logarithmic Frequency Scale

Since the frequency determinant resistors are all equal, and the cutting or corner frequency (εC) must be equal for a filter of the first, second, third or even fourth degree, such as the frequency determinant capacitors.

High-pass filters from the third and fourth order, as in filters of the first and second order, are created by simply changing the positions of the frequency-determining components (resistors and capacitors)in the equivalent low-pass filter. High-level filters can be designed by following the procedures we have seen in the filter and high-pass filter tutorials that have previously passed low. However, the total gain of high-order filters is constant, since all frequency-determining components are equal.

Filter Approaches

So far, we've looked at primary low- and high-pass filter circuits and the resulting frequency and phase responses. An ideal filter will give us maximum transition band gain and straightness, minimal stop band weakening, as well as a very steep transition band to stop band rolling(transition band).

Unsurprisingly, in linear analog filter design there are a number of "approach functions" that use a mathematical approach to get the best approach to the transfer function we need for filter design.

Such designs are known as Elliptical, Butterworth, Chebyshev, Bessel, Cauer and others. Of these five "classic" linear analogue filter approach functions, only the Butterworth filter and especially the low-passing Butterworth filter design will be considered the most commonly used function here.

Low Pass Butterworth Filter Design

The Frequency response of the Butterworth filter approach function is often referred to as the "maximum flat" (no fluctuation) response, since the transition band is designed to have a mathematically flat frequency response of 0hz (DC). Cutting frequency at -3db without fluctuation. Higher frequencies beyond the breakpoint are reduced to zero in the stop band at 20db/decade or 6db/octave. This is because it is only a "quality factor" of 0.707, "Q".

However, one of the main drawbacks of the Butterworth filter is that it achieves this transition band straightness at the expense of a wide transition band as the filter changes from the transition band to the stop band. It also has weak phase properties. The ideal Frequency response and standard Butterworth approaches, called "brick wall" filters for different filter orders, are given below.

Ideal Frequency Response for Butterworth Filter

Butterworth Filter
Ideal Frequency Response for Butterworth Filter

The higher the Butterworth filter sequence, the higher the number of cascading stages in the filter design. Then the filter becomes so close to the ideal "brick wall" response. In practice, however, Butterworth's ideal Frequency response cannot be achieved because it produces excessive band fluctuations.

Where there is a generalized equation representing the "n." order of the Butterworth filter, the Frequency response is given as follows:

Butterworth Filter

Where: n represents the filter sequence, equals Omega ω 2nε, and epsilon ε is the maximum transition band gain (Amax). If Amax is defined at a frequency equal to the cut-off-3db corner point (εc), the ε will then be equal to one, and therefore ε2 will be one. However, if you now want to define Amax at a different voltage gain value, such as 1dB or 1.1220 (1db = 20*logAmax), the new value of epsilon is ε as follows:

Butterworth Filter

Filter Design – Butterworth Low Passer

Find the order of an active low-pass Butterworth filter, the properties of which are given as follows: Amax at 200 radians/s (31.8 Hz) transition band frequency (wp) and Amin = -20dB at 0.5 dB and 800 radians/s stop band frequency (ws). Also design a suitable Butterworth filter circuit to meet these requirements:

First, the maximum passband gain Amax = 0.5 dB equals a gain of 1.0593, note that: 0.5 dB = 20 * log(a) at a frequency of 200 rads/s (wp), so epsilon ε value:

Butterworth Filter

Secondly, the minimum stop band gain is equal to a gain of 10 (- 20dB = 20*log (a)) at a stop band frequency (ws) of Amin = – 20db, 800 rads/s or 127.3 Hz.

Changing the values for the Frequency response of Butterworth filters in the general equation gives us the following:

Butterworth Filter

The next maximum value up to 2.42 is n = 3, because N must always be an integer ( integer). Therefore, "a tercum filter is required". To produce a tertering Butterworth filter, a quadrstatic filter phase with digits is required, along with a first-degree filter stage.

From the table of normalized low-pass Butterworth polynomials above, the coefficient for a terian filter is given as (1+s)(1+s+s2). This gives us a gain of 3-a = 1 or a = 2. Choosing a value for both feedback resistance Rf and resistance R1 in A = 1 + (Rf/R1) gives us values of 1kΩ and 1kω respectively: ( 1kΩ/1kΩ ) + 1 = 2.

We know that the cutting corner frequency, – 3dB point (wo) can be found using formula 1/CR, but then we need to find wo from wp transition band frequency,

Butterworth Filter
W0

Thus, the cutting corner frequency is given as 284 rads/s or 45.2 Hz, (284/2π) and using the familiar formula 1/CR we can find the values of resistors and capacitors for our third-degree circuit.

Butterworth Filter

Note that the preferred value closest to 0.352 uf will be 0.36 uF or 360nF.

Tercum butterworth low-pass filter

and finally, our butterworth filter circuit with a cutting angle frequency of 284 rads/s or 45.2 Hz, maximum transition band gain of 0.5 dB and minimum stop band gain of 20db is created as follows.

Butterworth Filter
Tercum butterworth low-pass filter

Therefore, our Butterworth low pass filter from 3rd degree will have a corner frequency of 45.2 Hz, C = 360nF and r = 10kω.