Filter types Butterworth low-pass filter Chebyshev type I low-pass filter Minimum filter order Moscow low-pass filter . Butterworth filters Defining the filter order

12.12.2022

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The frequency response of the Butterworth filter is described by the equation

Features of the Butterworth filter: nonlinear phase response; cutoff frequency independent of the number of poles; oscillatory nature of the transient response with a step input signal. As the filter order increases, the oscillatory nature increases.

Chebyshev filter

The frequency response of the Chebyshev filter is described by the equation

Where T n 2 (ω/ω n ) – Chebyshev polynomial n-th order.

The Chebyshev polynomial is calculated using the recurrent formula

Features of the Chebyshev filter: increased unevenness of the phase response; wave-like characteristic in the passband. The higher the coefficient of unevenness of the frequency response of the filter in the passband, the sharper the decline in the transition region at the same order. The transient oscillation of a stepped input signal is greater than that of a Butterworth filter. The quality factor of the Chebyshev filter poles is higher than that of the Butterworth filter.

Bessel filter

The frequency response of the Bessel filter is described by the equation

Where
;B n 2 (ω/ω cp h ) – Bessel polynomial n-th order.

The Bessel polynomial is calculated using the recurrent formula

Features of the Bessel filter: fairly uniform frequency response and phase response, approximated by the Gaussian function; the phase shift of the filter is proportional to the frequency, i.e. the filter has a frequency-independent group delay time. The cutoff frequency changes as the number of filter poles changes. The filter's frequency response is usually flatter than that of Butterworth and Chebyshev. This filter is especially suitable for pulse circuits and phase-sensitive signal processing.

Cauer filter (elliptical filter)

General view of the transfer function of the Cauer filter

Features of the Cauer filter: uneven frequency response in the passband and stopband; the sharpest drop in frequency response of all the above filters; implements the required transfer functions with a lower filter order than when using other types of filters.

Determining the filter order

The required filter order is determined by the formulas below and rounded to the nearest integer value. Butterworth filter order

Chebyshev filter order

For the Bessel filter, there is no formula for calculating the order; instead, tables are provided that correspond the filter order to the minimum required deviation of the delay time from unity at a given frequency and the loss level in dB).

When calculating the Bessel filter order, the following parameters are specified:

Permissible percentage deviation of group delay time at a given frequency ω ω cp h ;

The filter gain attenuation level can be set in dB at frequency ω , normalized relative to ω cp h .

Based on these data, the required order of the Bessel filter is determined.

Circuits of cascades of low-pass filters of the 1st and 2nd order

In Fig. 12.4, 12.5 show typical circuits of low-pass filter cascades.

A) b)

Rice. 12.4. Low-pass filter cascades of Butterworth, Chebyshev and Bessel: A - 1st order; b – 2nd order

A) b)

Rice. 12.5. Cauer low-pass filter cascades: A - 1st order; b – 2nd order

General view of the transfer functions of the Butterworth, Chebyshev and Bessel low-pass filters of the 1st and 2nd order

,
.

General view of the transfer functions of the Cauer low-pass filter of the 1st and 2nd order

,
.

The key difference between a 2nd order Cauer filter and a bandstop filter is that in the Cauer filter transfer function the frequency ratio Ω s ≠ 1.

Calculation method for Butterworth, Chebyshev and Bessel low-pass filters

This technique is based on the coefficients given in the tables and is valid for Butterworth, Chebyshev and Bessel filters. The method for calculating Cauer filters is given separately. The calculation of Butterworth, Chebyshev and Bessel low-pass filters begins with determining their order. For all filters, the minimum and maximum attenuation parameters and cutoff frequency are set. For Chebyshev filters, the coefficient of frequency response unevenness in the passband is additionally determined, and for Bessel filters, the group delay time is determined. Next, the transfer function of the filter is determined, which can be taken from the tables, and its 1st and 2nd order cascades are calculated, the following calculation procedure is observed:

Depending on the order and type of the filter, the circuits of its cascades are selected, while an even-order filter consists of n/2 2nd order cascades, and an odd order filter - from one 1st order cascade and ( n– 1)/2 cascades of the 2nd order;

To calculate a 1st order cascade:

The selected filter type and order determines the value b 1 1st order cascade;

By reducing the occupied area, the capacity rating is selected C and is calculated R according to the formula (you can also choose R, but it is recommended to choose C, for reasons of accuracy)

;

The gain is calculated TO at U 1 1st order cascade, which is determined from the relation

Where TO at U– gain of the filter as a whole; TO at U 2 , …, TO at Un– gain factors of 2nd order cascades;

To realize gain TO at U 1 it is necessary to set resistors based on the following relationship

R B = R A ּ (TO at U1 –1) .

To calculate a 2nd order cascade:

Reducing the occupied area, the nominal values of the containers are selected C 1 = C 2 = C;

Coefficients are selected from tables b 1 i And Q pi for 2nd order cascades;

According to a given capacitor rating C resistors are calculated R according to the formula

;

For the selected filter type, you must set the appropriate gain TO at Ui = 3 – (1/Q pi) of each 2nd order stage, by setting resistors based on the following relationship

R B = R A ּ (TO at Ui –1) ;

For Bessel filters, it is necessary to multiply the ratings of all capacitors by the required group delay time.

The frequency response of the Butterworth filter is described by the equation

Chebyshev filter

The frequency response of the Chebyshev filter is described by the equation

Where T n 2 (ω/ω n ) – Chebyshev polynomial n-th order.

The Chebyshev polynomial is calculated using the recurrent formula

Bessel filter

The frequency response of the Bessel filter is described by the equation

Where
;B n 2 (ω/ω cp h ) – Bessel polynomial n-th order.

The Bessel polynomial is calculated using the recurrent formula

Cauer filter (elliptical filter)

General view of the transfer function of the Cauer filter

Determining the filter order

The required filter order is determined by the formulas below and rounded to the nearest integer value. Butterworth filter order

Chebyshev filter order

When calculating the Bessel filter order, the following parameters are specified:

Permissible percentage deviation of group delay time at a given frequency ω ω cp h ;

The filter gain attenuation level can be set in dB at frequency ω , normalized relative to ω cp h .

Based on these data, the required order of the Bessel filter is determined.

Circuits of cascades of low-pass filters of the 1st and 2nd order

In Fig. 12.4, 12.5 show typical circuits of low-pass filter cascades.

A) b)

Rice. 12.4. Low-pass filter cascades of Butterworth, Chebyshev and Bessel: A - 1st order; b – 2nd order

A) b)

Rice. 12.5. Cauer low-pass filter cascades: A - 1st order; b – 2nd order

General view of the transfer functions of the Butterworth, Chebyshev and Bessel low-pass filters of the 1st and 2nd order

,
.

General view of the transfer functions of the Cauer low-pass filter of the 1st and 2nd order

,
.

The key difference between a 2nd order Cauer filter and a bandstop filter is that in the Cauer filter transfer function the frequency ratio Ω s ≠ 1.

Calculation method for Butterworth, Chebyshev and Bessel low-pass filters

To calculate a 1st order cascade:

The selected filter type and order determines the value b 1 1st order cascade;

By reducing the occupied area, the capacity rating is selected C and is calculated R according to the formula (you can also choose R, but it is recommended to choose C, for reasons of accuracy)

;

The gain is calculated TO at U 1 1st order cascade, which is determined from the relation

Where TO at U– gain of the filter as a whole; TO at U 2 , …, TO at Un– gain factors of 2nd order cascades;

To realize gain TO at U 1 it is necessary to set resistors based on the following relationship

R B = R A ּ (TO at U1 –1) .

To calculate a 2nd order cascade:

Reducing the occupied area, the nominal values of the containers are selected C 1 = C 2 = C;

Coefficients are selected from tables b 1 i And Q pi for 2nd order cascades;

According to a given capacitor rating C resistors are calculated R according to the formula

;

For the selected filter type, you must set the appropriate gain TO at Ui = 3 – (1/Q pi) of each 2nd order stage, by setting resistors based on the following relationship

R B = R A ּ (TO at Ui –1) ;

For Bessel filters, it is necessary to multiply the ratings of all capacitors by the required group delay time.

Butterworth filter

Butterworth Low Pass Filter Transfer Function n-order is characterized by the expression:

The amplitude-frequency response of the Butterworth filter has the following properties:

1) In any order n frequency response value

2) at the cutoff frequency u = u s

The frequency response of the low-pass filter decreases monotonically with increasing frequency. For this reason, Butterworth filters are called flat filters. Figure 3 shows graphs of the amplitude-frequency characteristics of Butterworth low-pass filters of 1-5 orders. Obviously, the higher the order of the filter, the more accurately the frequency response of an ideal low-pass filter is approximated.

Figure 3 - Frequency response for a low-pass Butterworth filter of order from 1 to 5

Figure 4 shows a circuit implementation of a Butterworth high-pass filter.

Figure 4 - Butterworth HPF-II

The advantage of the Butterworth filter is the smoothest frequency response at passband frequencies and its reduction to almost zero at stopband frequencies. The Butterworth filter is the only filter that retains the shape of the frequency response for higher orders (with the exception of a steeper roll-off of the characteristic at the suppression band), while many other types of filters (Bessel filter, Chebyshev filter, elliptic filter) have different shapes of the frequency response at different orders.

However, compared to Chebyshev filter types I and II or the elliptical filter, the Butterworth filter has a flatter rolloff and therefore must be of higher order (which is more difficult to implement) in order to provide the desired performance at the stopband frequencies.

Chebyshev filter

The squared modulus of the transfer function of the Chebyshev filter is determined by the expression:

where is the Chebyshev polynomial. The modulus of the transfer function of the Chebyshev filter is equal to unity at those frequencies where it becomes zero.

Chebyshev filters are usually used where it is necessary to use a small-order filter to provide the required frequency response characteristics, in particular, good suppression of frequencies from the suppression band, and the smoothness of the frequency response at the frequencies of the passband and suppression bands is not so important.

There are Chebyshev filters of types I and II.

Chebyshev filter of the first kind. This is a more common modification of Chebyshev filters. In the passband of such a filter, ripples are visible, the amplitude of which is determined by the ripple exponent e. In the case of an analog Chebyshev electronic filter, its order is equal to the number of reactive components used in its implementation. A steeper decline in the characteristic can be obtained by allowing ripples not only in the passband, but also in the suppression band, by adding zeros on the imaginary axis in the complex plane to the filter transfer function. This will, however, result in less effective suppression in the stopband. The resulting filter is an elliptic filter, also known as a Cauer filter.

The frequency response for a Chebyshev low-pass filter of the first type of the fourth order is presented in Figure 5.

Figure 5 - Frequency response for a Chebyshev low-pass filter of the first kind, fourth order

A type II Chebyshev filter (inverse Chebyshev filter) is used less frequently than a type I Chebyshev filter due to a less steep decline in the amplitude characteristic, which leads to an increase in the number of components. It has no ripple in the passband, but is present in the suppression band.

The frequency response for a Chebyshev low-pass filter of the second type of the fourth order is presented in Figure 6.

Figure 6 - Frequency response for a Chebyshev low-pass filter of type II

Figure 7 shows circuit implementations of Chebyshev high-pass filters of the 1st and 2nd order.

Figure 7 - Chebyshev high-pass filter: a) 1st order; b) II order

Properties of frequency characteristics of Chebyshev filters:

1) In the passband, the frequency response has an equal-wave character. On the interval (-1?sch?1) there is n points at which the function reaches a maximum value of 1 or a minimum value of . If n is odd, if n is even;

2) the value of the frequency response of the Chebyshev filter at the cutoff frequency is equal to

3) When the function decreases monotonically and tends to zero.

4) Parameter e determines the unevenness of the frequency response of the Chebyshev filter in the passband:

A comparison of the frequency response of Butterworth and Chebyshev filters shows that the Chebyshev filter provides greater attenuation in the passband than a Butterworth filter of the same order. The disadvantage of Chebyshev filters is that their phase-frequency characteristics in the passband differ significantly from linear ones.

For Butterworth and Chebyshev filters there are detailed tables that show the pole coordinates and coefficients of transfer functions of various orders.

In this article we will talk about the Butterworth filter, consider the orders of filters, decades and octaves, and analyze in detail the third-order Butterworth low-pass filter with calculation and circuit.

Introduction

In devices that use filters to shape the frequency spectrum of a signal, such as communications or control systems, the shape or width of the rolloff, also called the "cut-off band", for a simple first-order filter may be too long, or wide and active filters designed with more than one “order”. These types of filters are usually known as "high order" or "nth order" filters.

Filter order

The complexity or type of filter is determined by the "order" of the filters and depends on the number of reactive components such as capacitors or inductors in its design. We also know that the roll-off rate and therefore the transition bandwidth depends on the order number of the filter and that for a simple first-order filter it has a standard roll-off rate of 20 dB/decade or 6 dB/octave.

Then for a filter having the nth sequence number, it will have a subsequent roll-off rate of 20n dB/decade or 6n dB/octave. Thus:

first order filter has a decay rate of 20 dB/decade (6 dB/octave)
second order filter has a roll-off rate of 40 dB/decade (12 dB/octave)
fourth order filter has a roll-off frequency of 80 dB/decade (24 dB/octave), etc.

Higher order filters such as third, fourth and fifth order are usually formed by cascading together single first and second order filters.

For example, two second-order low-pass filters can be cascaded to produce a fourth-order low-pass filter, and so on. Although there is no limit to the order of a filter that can be formed, increasing the order increases its size and cost, and reduces its accuracy.

Decades and octaves

Last comment about Decades And Octaves. Frequency scale decade is a tenfold increase (multiplying by 10) or a tenfold decrease (dividing by 10). For example, 2 to 20 Hz represents one decade, whereas 50 to 5000 Hz represents two decades (50 to 500 Hz and then 500 to 5000 Hz).

Octave is a doubling (multiply by 2) or halving (dividing by 2) on the frequency scale. For example, 10 to 20 Hz represents one octave, and 2 to 16 Hz represents three octaves (2 to 4, 4 to 8, and finally 8 to 16 Hz), doubling the frequency each time. Anyway, logarithmic scales are widely used in the frequency domain to indicate frequency values when working with amplifiers and filters, so it is important to understand them.

Since the resistors that determine frequency are all equal, as are the capacitors that determine frequency, the cutoff or corner frequency (ƒ C) for a first, second, third, or even fourth order filter must also be equal and is found using the familiar equation:

As with first- and second-order filters, third- and fourth-order high-pass filters are formed by simply mutual exchange positions of frequency-determining components (resistors and capacitors) in an equivalent low-pass filter. High-order filters can be designed by following the procedures we saw earlier in the low-pass filter and high-pass filter tutorials. However, the overall gain of high order filters is fixed, since all components that determine frequency are the same.

Filter approximations

So far we have looked at low-pass and high-pass first-order filter circuits and their resulting frequency and phase characteristics. An ideal filter would give us the specifications of maximum passband gain and flatness, minimum passband attenuation, and a very steep passband to stop the roll-off (transition band), and so obviously a large number of network responses would satisfy these requirements.

It's no surprise that linear analog filter design has a number of "approximation functions" that use a mathematical approach to best approximate the transfer function we need to design filters.

Such designs are known as Elliptical, Butterworth, Chebyshev, Bessel, Cowher and many others. Of these five "classical" linear analog filter approximation functions, only Butterworth filter and especially the design low pass Butterworth filter will be considered here as its most commonly used function.

Low-pass Butterworth filter

Frequency response of the approximation function Butterworth filter also often called "as flat as possible" (ripple-free) response because the passband is designed to have a frequency response that is as flat as mathematically possible, from 0 Hz (DC) to the -3 dB cutoff frequency without ripple. Higher frequencies beyond the cutoff point are reduced to zero in the stopband at 20 dB/decade or 6 dB/octave. This is because it has a "quality factor", "Q" of only 0.707.

However, one of the main disadvantages of the Butterworth filter is that it achieves this passband flatness at the expense of a wide transition band as the filter changes from passband to stopband. It also has poor phase characteristics. The ideal frequency response, called a "brick wall" filter, and standard Butterworth approximations for various filter orders are given below.

Note that the higher the order of the Butterworth filter, the more quantity cascading steps in the filter design and the closer the filter comes to the ideal “brick wall” response.

However, in practice, the ideal Butterworth frequency response is unattainable because it causes excessive ripple in the passband.

Where the generalized equation representing the "nth" order Butterworth filter, the frequency response is given by:

Where: n represents the filter order, ω is 2πƒ, and ε is the maximum passband gain (A max).

If A max is defined at a frequency equal to the -3 dB corner cutoff point (ƒc), then ε will be equal to one and therefore ε 2 will also be equal to one. However, if you now want to determine A max at a different voltage gain value, for example 1 dB or 1.1220 (1 dB = 20 * logA max), then the new value of ε is found using the formula:

Substituting the data into the equations, we get:

Frequency response filter can be determined mathematically by its transfer function with the voltage transfer standard Function H (jω) and is written as:

Note: (jω) can also be written as (s) to indicate S-regions. and the resulting transfer function for the second order low pass filter is given by:

Normalized low-pass Butterworth filter polynomials

To aid in the design of his low-pass filters, Butterworth created standard tables of normalized second-order low-pass polynomials given coefficient values that correspond to an angle cutoff frequency of 1 radian/s.

N	Normalized denominator polynomials in factored form
1	(1+S)
2	(1 + 1.414 s + s 2)
3	(1 + s) (1 + s + s 2)
4	(1 + 0.765 s + s 2) (1 + 1.848 s + s 2)
5	(1 + s) (1 + 0.618 s + s 2) (1 + 1.618 s + s 2)
6	(1 + 0.518 s + s 2) (1 + 1.414 s + s 2) (1 + 1.932 s + s 2)
7	(1 + s) (1 + 0.445 s + s 2) (1 + 1.247 s + s 2) (1 + 1.802 s + s 2)
8	(1 + 0.390 s + s 2) (1 + 1.111 s + s 2) (1 + 1.663 s + s 2) (1 + 1.962 s + s 2)
9	(1 + s) (1 + 0.347 s + s 2) (1 + s + s 2) (1 + 1.532 s + s 2) (1 + 1.879 s + s 2)
10	(1 + 0.313 s + s 2) (1 + 0.908 s + s 2) (1 + 1.414 s + s 2) (1 + 1.782 s + s 2) (1 + 1.975 s + s 2)

Calculation and circuit of the low-pass Butterworth filter

Find the order of an active low-pass Butterworth filter whose characteristics are given as: A max = 0.5 dB at a passband frequency (ωp) of 200 radians/sec (31.8 Hz), and A min = -20 dB at a stopband frequency (ωs ) 800 radians/sec. Also design a suitable Butterworth filter circuit to meet these requirements.

Firstly, the maximum passband gain A max = 0.5 dB, which is equal to the gain 1,0593 , remember that: 0.5 dB = 20 * log(A) at a frequency (ωp) of 200 rad/s, so the value of epsilon ε is found by:

Secondly, the minimum stop band gain A min = -20 dB, which is equal to the gain 10 (-20 dB = 20 * log(A)) at a stop band frequency (ωs) of 800 rad/s or 127.3 Hz.

Substituting the values into the general equation for the frequency response of Butterworth filters gives us the following:

Since n must always be an integer, the next highest value of 2.42 would be n = 3, so "a third order filter is required" and to create Butterworth filter third order, the second order filter stage requires cascading with the first order filter stage.

From the above table of normalized low pass Butterworth polynomials, the coefficient for a third order filter is given as (1 + s)(1 + s + s 2) and this gives us a gain of 3-A = 1 or A = 2. V A = 1 + (Rf / R1), choosing a value as for a resistor feedback Rf and resistor R1 gives us the values of 1 kOhm and 1 kOhm, respectively, as: (1 kOhm / 1 kOhm) + 1 = 2.

We know that the corner frequency cutoff, -3 dB point (ω o) can be found using the formula 1/CR, but we need to find ω o from the passband frequency ω p.

So the angle cutoff frequency is given as 284 rad/s or 45.2 Hz (284/2π), and using the familiar 1/RC formula we can find the resistor and capacitor values for our third order circuit.

Note that the closest preferred value up to 0.352 µF would be 0.36 µF or 360 nF.

And finally, our low-pass filter circuit Butterworth third order with an angular cutoff frequency of 284 rad/s or 45.2 Hz, a maximum passband gain of 0.5 dB and a minimum stopband gain of 20 dB is constructed as follows.

So for our 3rd order Butterworth low pass filter with a corner frequency of 45.2 Hz, C = 360 nF and R = 10 kΩ

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

4th order Butterworth filter

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Chebyshev filter 3rd order

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Chebyshev filter 4 orders

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Bessel filter 3rd order

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Bessel filter 4th order

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Analyze the influence of errors in setting the digital low-pass filter coefficients on the frequency response (by changing one of the coefficients b j).

Describe the nature of the change in frequency response. Draw a conclusion about the effect of changing one of the coefficients on the behavior of the filter.

We will analyze the influence of errors in setting the digital low-pass filter coefficients on the frequency response using the example of a 4th order Bessel filter.

Let us choose the deviation of the coefficients ε equal to –1.5%, so that the maximum deviation of the frequency response is about 10%.

The frequency response of an “ideal” filter and filters with changed coefficients by the value ε is shown in the figure:

AND

The figure shows that the greatest influence on the frequency response is exerted by changes in the coefficients b 1 and b 2 (their value exceeds the value of other coefficients). Using a negative value of ε, we note that positive coefficients reduce the amplitude in the lower part of the spectrum, while negative coefficients increase it. For a positive value of ε, everything happens the other way around.

Quantize the digital filter coefficients by such a number of binary digits that the maximum deviation of the frequency response from the original is about 10 - 20%. Sketch the frequency response and describe the nature of its change. b j By changing the number of digits of the fractional part of the coefficients

Note that the maximum deviation of the frequency response from the original one does not exceed 20% is obtained when n≥3. n Type of frequency response at different

n =3, maximum frequency response deviation =19.7%

n =4, maximum frequency response deviation =13.2%

n =5, maximum frequency response deviation =5.8%

n =6, maximum frequency response deviation =1.7%

Thus, it can be noted that increasing the bit depth when quantizing filter coefficients leads to the fact that the frequency response of the filter tends more and more to the original one. However, it should be noted that this complicates the physical realizability of the filter.

Quantization at different n can be seen in the figure: