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

Features of the Butterworth filter: non-linear phase response; cutoff frequency independent of the number of poles; oscillatory character of the transient response with a step input signal. As the filter order increases, the oscillatory character is enhanced.

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 recursive formula

Features of the Chebyshev filter: increased non-uniformity of the phase response; undulating characteristic in the passband. The higher the coefficient of non-uniformity of the frequency response of the filter in the passband, the sharper the drop in the transition region with the same order. The transient ripple with a stepped input signal is stronger than that of a Butterworth filter. The Q-factor of the poles of the Chebyshev filter 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 s ) - 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 rolloff is usually more shallow than that of Butterworth and Chebyshev. This filter is especially well suited for impulse circuits and phase-sensitive signal processing.

Cauer filter (elliptical filter)

General view of the transfer function of the Cauer filter

.

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

Defining filter order

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

.

Chebyshev filter order

.

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

When calculating the order of the Bessel filter, the following parameters are set:

Percentage tolerance of group delay time at a given frequency ω ω cp s ;

The attenuation level of the filter gain can be set in dB at the frequency ω normalized with respect to ω cp s .

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

Schemes of cascades of high-pressure filters of the 1st and 2nd order

In fig. 12.4, 12.5 show typical schemes of LPF stages.

a) b)

Rice. 12.4. LPF stages of Butterworth, Chebyshev and Bessel: a - 1st order; b - 2nd order

a) b)

Rice. 12.5. Cauer LPF cascades: a - 1st order; b - 2nd order

General view of transfer functions of LPF Butterworth, Chebyshev and Bessel 1st and 2nd order

,
.

General view of the transfer functions of the Cauer LPF of the 1st and 2nd order

,
.

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

Methodology for calculating LPFs for Butterworth, Chebyshev and Bessel

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

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

To calculate the 1st order cascade:

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

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

;

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

,

where TO at U- gain of the filter as a whole; TO at U 2 , …, TO at Un- amplification factors of the 2nd order stages;

To implement amplification TO at U 1 it is necessary to set the resistors based on the following ratio

R B = R A ּ (TO at U1 –1) .

To calculate the 2nd order cascade:

Reducing the occupied area, the ratings of the capacities are selected C 1 = C 2 = C;

Coefficients are selected according to tables b 1 i and Q pi for cascades of the 2nd order;

At 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 stage of the 2nd order, by setting the resistors, based on the following ratio

R B = R A ּ (TO at Ui –1) ;

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

Much of the theory behind digital IIR filters (ie, infinite impulse response filters) requires an understanding of how to compute continuous-time filters. Therefore, this section will provide design formulas for several standard types of analog filters, including Butterworth, Bessel and Chebyshev type I and II filters. A detailed analysis of the advantages and disadvantages of methods for approximating the given characteristics corresponding to these filters can be found in a number of works devoted to methods for calculating analog filters, therefore, below we will only briefly list the main properties of filters of each type and give the calculated ratios necessary to obtain the coefficients of analog filters.

Let it be necessary to calculate a normalized low-pass filter with a cutoff frequency equal to Ω = 1 rad / s. As a rule, the square of the amplitude characteristic will be used as an approximated function (the Bessel filter is an exception). We will assume that the transfer function of the analog filter is a rational function of the variable S of the following form:

Low-pass Butterworth filters are characterized by having the smoothest possible amplitude response at the origin in the s-plane. This means that all existing derivatives of the amplitude characteristic at the origin are equal to zero. The squared amplitude characteristic of the normalized (i.e., having a cutoff frequency of 1 rad / s) Butterworth filter is:

where n - filter order. Analytically extending function (14.2) to the entire S-plane, we obtain

All poles (14.3) are located on the unit circle at the same distance from each other in S-plane ... Let us express the transfer function H (s) through the poles located in the left half-plane S :

Where (14.4)

Where k = 1,2 ... ..n (14.5)

a k 0 is the constant of normalization. Using formulas (14.2) and (14.5), several properties of low-pass Butterworth filters can be formulated.

Low Pass Butterworth Filters Properties:

1. Butterworth filters have only poles (all zeros of the transfer functions of these filters are located at infinity).

2. At a frequency of Ω = 1 rad / s, the transmission coefficient of the Butterworth filters is (i.e., at the cutoff frequency, their amplitude characteristic decreases by 3 dB).

3. Filter order n completely defines the entire filter. In practice, the order of the Butterworth filter is usually calculated from the condition of providing a certain attenuation at some given frequency Ω t> 1. The order of the filter providing at the frequency Ω = Ω t< уровень амплитудной характеристики, равный 1/А, можно найти из соотношения

Rice. 14.1. Pole locations for the analog low-pass Butterworth filter.

Rice. 14.2- Amplitude and phase characteristics, as well as the characteristic of the group delay of the analog low-pass Butterworth filter.

Let, for example, required at frequency Ω t = 2 rad / s provide an attenuation equal to A = 100. Then

Rounded n up to an integer, we find that the given attenuation will provide the 7th order Butterworth filter.

Solution... Using as design characteristics 1 / A == 0.0005 (which corresponds to an attenuation of 66 dB) and Ω t = 2, we get n== 10.97. Rounding gives n = 11... In fig. 14.1 shows the location of the poles of the calculated Butterworth filter in s-plane... The amplitude (on a logarithmic scale) and phase characteristics, as well as the characteristic of the group delay of this filter are shown in Fig. 14.2.

Butterworth filter

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

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

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

Figure 4 shows the circuit implementation of the Butterworth HPF.

Figure 4 - Butterworth HPF-II

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

However, compared to Type I and II Chebyshev filters or an elliptical filter, the Butterworth filter has a shallower roll-off and therefore must have a higher order (which is more difficult to implement) in order to provide the desired characteristics at the stopband frequencies.

Chebyshev filter

The square of the 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 one at those frequencies where it vanishes.

Chebyshev filters are usually used where it is required to provide the required frequency response characteristics using a small order filter, 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 I and II types.

Chebyshev filter of the 1st 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 index e. In the case of an analog electronic Chebyshev filter, its order is equal to the number of reactive components used in its implementation. A steeper roll-off of the characteristic can be obtained if ripple is allowed not only in the passband, but also in the suppression band, by adding zeros to the filter transfer function on the imaginary axis jsh in the complex plane. This, however, will result in less effective suppression in the notch band. The resulting filter is an elliptical filter, also known as a Cauer filter.

The frequency response for the Chebyshev low-pass filter of the first kind of the fourth order is shown in Figure 5.

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

The Chebyshev type II filter (inverse Chebyshev filter) is used less frequently than the Chebyshev type I filter due to the less steep roll-off of 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 the Chebyshev low-pass filter of the second kind of the fourth order is shown in Figure 6.

Figure 6 - AFC for the Chebyshev low-pass filter of the II kind

Figure 7 shows the schematic realizations of the Chebyshev HPFs of the I and II order.

Figure 7 - HPF Chebyshev: a) I order; b) II order

Frequency response properties of Chebyshev filters:

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

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

3) At, 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:

Comparison of the frequency response of the Butterworth and Chebyshev filters shows that the Chebyshev filter provides more attenuation in the passband than the 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 coordinates of the poles and the coefficients of the transfer functions of various orders.

When analyzing filters and calculating their parameters, some standard terms are always used and it makes sense to stick to them from the very beginning.

Suppose you want a low pass filter with a flat passband response and a sharp transition to notch band. The final slope in the stopband will always be 6n dB / octave, where n is the number of poles. One capacitor (or inductor) is required per pole, so the requirements for the final roll-off rate of the filter's frequency response, roughly speaking, determine its complexity.

Now, suppose you decide to use a 6-pole low-pass filter. You are guaranteed a final rolloff of 36 dB / octave at high frequencies. In turn, it is now possible to optimize the filter circuit in the sense of providing the most flat response in the passband by reducing the slope of the transition from passband to stopband. On the other hand, by allowing some passband ripple, a steeper transition from passband to stopband can be achieved. The third criterion, which may be important, describes the ability of the filter to pass signals with a spectrum lying in the passband without distorting their shape caused by phase shifts. You can also be interested in rise time, overshoot and settling time.

There are known filter design techniques suitable for optimizing any of these characteristics or combinations thereof. Really smart filter choices don't work as described above; As a rule, the required uniformity of the characteristics in the passband and the necessary attenuation at a certain frequency outside the passband and other parameters are set first. After that, the most suitable circuit is selected with the number of poles sufficient to satisfy all these requirements. The next few sections will look at the three most popular filter types, namely Butterworth (the flattest possible passband response), the Chebyshev filter (the steepest pass-to-pass transition), and the Bessel filter (the flattest lag response). Any of these types of filters can be implemented using various filter schemes; we will discuss some of them later. All of them are equally suitable for constructing low-pass, high-pass and band-pass filters.

Butterworth and Chebyshev filters. The Butterworth filter provides the flattest response in the passband, which is achieved at the cost of smooth response in the transition region, i.e. between bandwidth and delay. As will be shown later, it also has a poor phase-frequency response. Its frequency response is given by the following formula:
U out / U in = 1 / 1/2,
where n is the order of the filter (number of poles). Increasing the number of poles makes it possible to flatten the passband response and increase the slope from passband to suppression bandwidth, as shown in Fig. 5.10.

Rice. 5.10 Normalized characteristics of low-pass Butterworth filters. Note the increase in roll-off with increasing filter order.

Choosing a Butterworth filter, we compromise everything else for the sake of the flattest possible response. Its characteristic goes horizontally, starting from zero frequency, its inflection begins at a cutoff frequency ƒ s - this frequency usually corresponds to the point -3 dB.

In most applications, the most significant factor is that passband ripple should not exceed some specific value, say 1 dB. The Chebyshev filter meets this requirement, while some unevenness of the characteristic is allowed in the entire passband, but at the same time the sharpness of its kink is greatly increased. For the Chebyshev filter, the number of poles and the unevenness in the passband are set. Allowing for increased passband ripple results in a sharper kink. The frequency response of this filter is given by the following relationship
U out / U in = 1 / 1/2,
where C n is a Chebyshev polynomial of the first kind of degree n, and ε is a constant that determines the non-uniformity of the characteristic in the passband. The Chebyshev filter, like the Butterworth filter, has phase-frequency characteristics that are far from ideal. In fig. 5.11 shows for comparison the characteristics of 6-pole low-pass Chebyshev and Butterworth filters. As you can easily see, both are much better than a 6-pole RC filter.

Rice. 5.11. Comparison of the characteristics of some commonly used 6-pole low-pass filters. The characteristics of the same filters are shown in both logarithmic (top) and linear (bottom) scales. 1 - Bessel filter; 2 - Butterworth filter; 3 - Chebyshev filter (ripple 0.5 dB).

In fact, a Butterworth filter with a maximally flat response in the passband is not as attractive as it might seem, since in any case you have to put up with some unevenness in the passband (for a Butterworth filter, this will be a gradual decrease in the characteristic when approaching the frequency ƒ s, and for the Chebyshev filter - ripple distributed over the entire passband). In addition, active filters built from elements whose ratings have a certain tolerance will have a characteristic that differs from the calculated one, which means that in reality there will always be some unevenness in the passband on the characteristic of the Butterworth filter. In fig. 5.12 illustrates the influence of the most undesirable deviations in the values ​​of capacitor capacitance and resistor resistance on the filter characteristic.

Rice. 5.12. The influence of changes in the parameters of the elements on the characteristic of the active filter.

In light of the above, a very rational structure is the Chebyshev filter. Sometimes it is called an equal-wave filter, since its characteristic in the transition region has a high steepness due to the fact that several equal-sized ripples are distributed over the passband, the number of which increases with the order of the filter. Even with relatively low ripple (about 0.1 dB), the Chebyshev filter provides a much higher slope in the transition region than the Butterworth filter. To quantify this difference, assume that you need a filter with no more than 0.1 dB passband ripple and 20 dB attenuation at a frequency 25% different from the passband cutoff frequency. Calculation shows that in this case a 19-pole Butterworth filter or just an 8-pole Chebyshev filter is required.

The idea that it is possible to put up with the ripple of the characteristic in the passband in order to increase the steepness of the transition section is brought to its logical conclusion in the idea of ​​the so-called elliptical filter (or Cauer filter), in which ripple of the characteristic both in the passband and in the band is allowed delay for the sake of ensuring the steepness of the transition section is even greater than that of the Chebyshev filter characteristic. With the help of a computer, elliptical filters can be constructed as easily as the classical Chebyshev and Butterworth filters. In fig. 5.13 shows the graphical setting of the frequency response of the filter. In this case (low-pass filter), the allowable range of the filter gain (i.e., flatness) in the passband, the minimum frequency at which the characteristic leaves the passband, the maximum frequency where the characteristic passes into the stopband, and the minimum attenuation in the passband are determined. detention.

Rice. 5.13. Setting the parameters of the frequency response of the filter.

Bessel filters. As stated earlier, the frequency response of the filter does not provide complete information about it. A filter with a flat frequency response can have large phase shifts. As a result, the waveform, the spectrum of which lies in the passband, will be distorted when passing through the filter. In a situation where the waveform is of paramount importance, it is desirable to have a linear phase filter (constant time lag filter) available. Requiring the filter to provide a linear phase shift versus frequency is equivalent to requiring a constant latency for a signal whose spectrum is in the passband, i.e., no distortion of the waveform. A Bessel filter (also called a Thomson filter) has the flattest portion of the lag time in the passband, just as a Butterworth filter has the flattest frequency response. To understand what kind of improvement in the time domain the Bessel filter gives, take a look at Fig. 5.14, which shows frequency-normalized lag time plots for 6-pole Bessel and Butterworth low-pass filters. Poor lag time characteristics of the Butterworth filter cause overshoot effects when passing through the filter pulse signals. On the other hand, for the constancy of the delay times of the Bessel filter, one has to pay the price of the fact that its amplitude-frequency characteristic has an even flatter transition section between the passband and stopband than even the characteristic of the Butterworth filter.

Rice. 5.14. Comparison of time delays for 6-band low-pass Bessel (1) and Butterworth (2) filters. The Bessel filter, due to its excellent properties in the time domain, gives the least distortion of the waveform.

There are many different filter design techniques that attempt to improve the performance of a Bessel filter in the time domain, sacrificing in part lag time consistency in order to reduce rise time and improve frequency response. The Gaussian filter has almost as good phase response as the Bessel filter, but with improved transient response. Another interesting class is filters that make it possible to achieve the same ripple in the delay time curve in the passband (similar to the ripple of the frequency response of the Chebyshev filter) and provide approximately the same delay for signals with a spectrum up to the stopband. Another approach to creating filters with constant lag is the use of all-pass filters, also called time-domain equalizers. These filters have a constant frequency response, and the phase shift can be varied according to specific requirements. Thus, they can be used to equalize the lag time of any filters, in particular Butterworth and Chebyshev filters.

Comparison of filters. Despite the previous remarks about the transient response of Bessel filters, it still has very good properties in the time domain compared to Butterworth and Chebyshev filters. The Chebyshev filter itself, with its very suitable frequency response, has the worst time-domain parameters of all three types of filters. The Butterworth filter offers a trade-off between frequency and timing. In fig. 5.15 provides information on the performance characteristics of these three filter types in the time domain, in addition to the frequency response plots previously shown. Based on this data, it can be concluded that in cases where the filter parameters in the time domain are important, it is desirable to apply the Bessel filter.

Rice. 5.15. Comparison of transients of 6-pole low-pass filters. The curves are normalized by converting the 3 dB attenuation to a frequency of 1 Hz. 1 - Bessel filter; 2 - Butterworth filter; 3 - Chebyshev filter (ripple 0.5 dB).

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

Features of the Butterworth filter: non-linear phase response; cutoff frequency independent of the number of poles; oscillatory character of the transient response with a step input signal. As the filter order increases, the oscillatory character is enhanced.

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 recursive formula

Features of the Chebyshev filter: increased non-uniformity of the phase response; undulating characteristic in the passband. The higher the coefficient of non-uniformity of the frequency response of the filter in the passband, the sharper the drop in the transition region with the same order. The transient ripple with a stepped input signal is stronger than that of a Butterworth filter. The Q-factor of the poles of the Chebyshev filter 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 s ) - 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 rolloff is usually more shallow than that of Butterworth and Chebyshev. This filter is especially well suited for impulse circuits and phase-sensitive signal processing.

Cauer filter (elliptical filter)

General view of the transfer function of the Cauer filter

.

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

Defining filter order

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

.

Chebyshev filter order

.

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

When calculating the order of the Bessel filter, the following parameters are set:

Percentage tolerance of group delay time at a given frequency ω ω cp s ;

The attenuation level of the filter gain can be set in dB at the frequency ω normalized with respect to ω cp s .

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

Schemes of cascades of high-pressure filters of the 1st and 2nd order

In fig. 12.4, 12.5 show typical schemes of LPF stages.

a) b)

Rice. 12.4. LPF stages of Butterworth, Chebyshev and Bessel: a - 1st order; b - 2nd order

a) b)

Rice. 12.5. Cauer LPF cascades: a - 1st order; b - 2nd order

General view of transfer functions of LPF Butterworth, Chebyshev and Bessel 1st and 2nd order

,
.

General view of the transfer functions of the Cauer LPF of the 1st and 2nd order

,
.

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

Methodology for calculating LPFs for Butterworth, Chebyshev and Bessel

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

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

To calculate the 1st order cascade:

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

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

;

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

,

where TO at U- gain of the filter as a whole; TO at U 2 , …, TO at Un- amplification factors of the 2nd order stages;

To implement amplification TO at U 1 it is necessary to set the resistors based on the following ratio

R B = R A ּ (TO at U1 –1) .

To calculate the 2nd order cascade:

Reducing the occupied area, the ratings of the capacities are selected C 1 = C 2 = C;

Coefficients are selected according to tables b 1 i and Q pi for cascades of the 2nd order;

At 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 stage of the 2nd order, by setting the resistors, based on the following ratio

R B = R A ּ (TO at Ui –1) ;

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