1 Determine the filter order. The filter order is the number of reactive elements in the low-pass filter and high-pass filter.
Where 
- Butterworth function corresponding to the permissible frequency 
.
- permissible attenuation.
2 We draw a filter circuit of the resulting order. In practical implementation, circuits with fewer inductances are preferable.
3 We calculate the constant transformations of the filter.
, mH
, nF
4 For an ideal filter with a generator resistance of 1 Ohm, a load resistance of 1 Ohm, 
A table of normalized Butterworth filter coefficients has been compiled. In each row of the table, the coefficients are symmetrical, increasing toward the middle and then decreasing.
5 To find the elements of the circuit, it is necessary to multiply the constant transformations by the coefficient from the table.
|
Filter order |
Filter sequence numbers m |
|||||||||
Calculate the parameters of the Butterworth low-pass filter if PP=0.15 kHz, 
=25 kHz, 
=30 dB, 
=75 Ohm. Find 
for three points.
29.3 Butterworth HPF.
High-pass filters are four-terminal networks, which have a range of ( 
) the attenuation is small, and in the range ( 
) is large, that is, the filter must pass high-frequency currents into the load.
Since the high-pass filter must pass high-frequency currents, in the path of the current going to the load there must be a frequency-dependent element that passes high-frequency currents well and poorly low-frequency currents. Such an element is a capacitor.
F 
HF T-shape
U-shaped high-pass filter
The capacitor is placed in series with the load, since 
and with increasing frequency 
decreases, therefore high-frequency currents easily pass to the load through the capacitor. The inductor is placed parallel to the load, since 
and increases with increasing frequency 
, therefore, low-frequency currents are closed through inductances and will not enter the load.
The calculation of the Butterworth low-pass filter is similar to the calculation of the Butterworth low-pass filter; it is carried out using the same formulas, only
.
Calculate the Butterworth high-pass filter if 
Om, 
kHz, 
dB, 
kHz. Find: 
.
Lesson topic 30: Band-pass and notch filters Butterworth.
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.
Much of the theory behind designing digital IIR filters (i.e., infinite impulse response filters) requires an understanding of methods for designing continuous-time filters. Therefore, this section will provide calculation 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 given characteristics corresponding to these filters can be found in a number of works devoted to methods for calculating analog filters, so below we will only briefly list the main properties of filters of each type and provide the calculated relationships necessary to obtain the coefficients of analog filters.
Suppose we need 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 the 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 to the origin are equal to zero. The squared amplitude response of a normalized (i.e., having a cutoff frequency of 1 rad/s) Butterworth filter is equal to:
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 N(s) through the poles located in the left half-plane S :
Where (14.4)
Where k =1.2…..n (14.5)
A k 0 - normalization constant. Using formulas (14.2) and (14.5), several properties of low-pass Butterworth filters can be formulated.
Properties of low-pass Butterworth filters:
1. Butterworth filters have only poles (all zeros of the transfer functions of these filters are located at infinity).
2. At a frequency Ω=1 rad/s, the transmission coefficient of Butterworth filters is equal (i.e., at the cutoff frequency, their amplitude characteristic drops 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 ensuring a certain attenuation at a certain given frequency Ω t > 1. The order of the filter, providing at frequency Ω = Ω t< уровень амплитудной характеристики, равный 1/А, можно найти из соотношения
Rice. 14.1. Analog low pass Butterworth filter pole locations.
Rice. 14.2- Amplitude and phase characteristics, as well as group delay characteristics of an analog low-pass Butterworth filter.
Let, for example, required at frequency Ω t = 2 rad/s provide attenuation equal to A = 100. Then
Rounded n upward to an integer, we find that the given attenuation will be provided by a 7th order Butterworth filter.
Solution. Using 1/A == 0.0005 (corresponding to 66 dB attenuation) and Ω t = 2, we get n== 10.97. Rounding gives n=11. In Fig. Figure 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 group delay characteristic of this filter are presented in Fig. 14.2.
In filters, the calculation usually begins with setting the filter parameters, the most important of which is the frequency response. As we already discussed in the article, first the requirements of a given filter are brought to the requirements of the low-pass filter prototype. An example of the requirements for the amplitude-frequency response of a low-pass filter prototype of the designed filter is shown in Figure 1.
Figure 1. Example of the normalized amplitude-frequency response of a low-pass filter
This graph shows the dependence of the filter transmission coefficient on the normalized frequency ξ , Where ξ = f/f V
The graph shown in Figure 1 shows that the permissible unevenness of the transmission coefficient is specified in the passband. In the stopband, the minimum coefficient of suppression of the interfering signal is set. The real filter can have any shape. The main thing is that it does not cross the boundaries of the specified requirements.
For quite a long time, the filter was calculated by selecting the amplitude-frequency response using standard links (m-link or k-link). Similar method called the application method. It was quite complicated and did not provide the optimal ratio of the quality of the developed filter and the number of links. Therefore, mathematical methods have been developed to approximate the amplitude-frequency response with given characteristics.
In mathematics, approximation is the representation of a complex relationship by some known function. Usually this function is quite simple. When developing a filter, it is important that the approximating function can be easily implemented in circuitry. To do this, the functions are implemented using the zeros and poles of the transmission coefficient of a four-port network, in this case a filter. They are easily implemented using LC circuits or feedback loops.
The most common type of approximation of the frequency response of a filter is the Butterworth approximation. Such filters are called Butterworth filters.
Butterworth filters
A distinctive feature of the amplitude-frequency response of the Butterworth filter is the absence of minima and maxima in the passband and stopband. The frequency response rolloff at the edge of the passband of these filters is 3 dB. If a filter is required to have a lower ripple value in the passband, then the correct filter frequency f in is selected above the specified upper frequency of the passband. The frequency response approximation function for the low-pass filter prototype of the Butterworth filter is as follows:
(1),
Where ξ
— normalized frequency;
n— filter order.
In this case, the real amplitude-frequency characteristic of the filter being developed can be obtained by multiplying the normalized frequency ξ to the filter cutoff frequency. For a low-pass Butterworth filter, the frequency response approximation function will look like this:
(2).
Now let us note that when calculating filters, the concept of a complex s-plane is widely used, on which the circular frequency is plotted along the ordinate axis jω, and along the x-axis is the reciprocal of the quality factor. In this way, it is possible to determine the main parameters of the LC circuits that are part of the filter circuit: tuning frequency (resonant frequency) and quality factor. The transition to the s-plane is carried out using .
A detailed derivation of the pole positions of the Butterworth filter on the complex s-plane is given in. The main thing for us is that the poles of this filter are located on the unit circle at an equal distance from each other. The number of poles is determined by the order of the filter.
Figure 2 shows the pole locations for a first order Butterworth filter. The frequency response corresponding to a given arrangement of poles on the complex s-plane is shown nearby.
Figure 2. Pole location and frequency response of a first order Butterworth filter
Figure 2 shows that for a first-order filter, the pole must be tuned to zero frequency and its quality factor must be equal to unity. The frequency response graph shows that the tuning frequency of the pole is indeed zero, and the quality factor of the pole is such that at the cutoff frequency of the normalized Butterworth filter, equal to unity, its transmission coefficient is −3 dB.
The poles for the second order Butterworth filter are determined in exactly the same way. This time, the pole tuning frequency is selected at the intersection of the unit circle with a straight line passing through the center of the circle at an angle of 45°. An example of the location of the poles on the complex s-plane and the frequency response of a second-order Butterworth filter is shown in Figure 3.
Figure 3. Pole location and frequency response of a second-order Butterworth filter
In this case, the resonant frequency of the pole is located close to the cutoff frequency of the normalized filter. It is equal to 0.707. The pole quality factor according to the pole location graph is the root of two times higher than the pole quality factor of a first-order Butterworth filter, so the slope of the amplitude-frequency response is greater. (Pay attention to the numbers on the right side of the graph. With a frequency detuning of 2, the suppression is already 13 dB) The left side of the amplitude-frequency response of the pole turns out to be flat. This is due to the influence of the pole located in the negative frequency zone.
The location of the poles and the amplitude-frequency response of the third-order Butterworth filter is shown in Figure 4.
Figure 4. Third order Butterworth filter pole arrangement
As can be seen from the graphs shown in Figures 2...5, as the order of the Butterworth filter increases, the slope of the amplitude-frequency response increases and the required quality factor of the second-order circuit (circuit) that implements the pole of the filter’s transmission characteristic increases. It is the increase in the required quality factor that limits the maximum order of the filter that can be implemented. Currently, it is possible to implement Butterworth filters up to the eighth - tenth order.
Chebyshev filters
In Chebyshev filters, the amplitude-frequency response is approximated as follows:
(3),
In this case, the amplitude-frequency response of a real Chebyshev filter, just like in the Butterworth filter, can be obtained by multiplying the normalized frequency ξ to the cutoff frequency of the filter being developed. For a low-pass Chebyshev filter, the amplitude-frequency response can be determined as follows:
(4).
The amplitude-frequency response of the low-pass Chebyshev filter is characterized by a steeper decline in the frequency range above the upper pass frequency. This gain is achieved due to the appearance of frequency response unevenness in the passband. The unevenness of the approximation function of the frequency response of the Chebyshev filter is caused by the higher quality factor of the poles.
A detailed derivation of the position of the poles of the approximating function of the Chebyshev filter on the s-plane is given in. What is important for us is that the poles of the Chebyshev filter are located on an ellipse, the major axis of which coincides with the axis of normalized frequencies. On this axis, the ellipse passes through the cutoff frequency point of the low-pass filter.
In the normalized version, this point is equal to one. The second axis is determined by the unevenness of the frequency response approximation function in the passband. The greater the permissible ripple in the passband, the smaller this axis. There is a kind of “flattening” of the unit circle of the Butterworth filter. The poles seem to be approaching the frequency axis. This corresponds to an increase in the quality factor of the filter poles. The greater the unevenness in the passband, the greater the quality factor of the poles, the greater the rate of increase in attenuation in the stopband of the Chebyshev filter. The number of poles of the frequency response approximation function is determined by the order of the Chebyshev filter.
It should be noted that there is no first order Chebyshev filter. The location of the poles and frequency response of the second-order Chebyshev filter is shown in Figure 5. The characteristic of the Chebyshev filter is interesting in that the frequencies of the poles are clearly visible on it. They correspond to the maximum frequency response in the passband. For a second-order filter, the pole frequency corresponds to ξ =0.707.
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 the stopband. The final slope of the response in the stopband will always be 6n dB/octave, where n is the number of “poles”. One capacitor (or inductor) is needed per pole, so the final roll-off rate requirements of the filter roughly determine its complexity.
Now let's say you decide to use a 6-pole low-pass filter. You are guaranteed a final decline in performance by high frequencies 36 dB/octave. In turn, it is now possible to optimize the filter design in the sense of providing the most flat response in the passband by reducing the slope of the transition from the passband to the stopband. On the other hand, by allowing some ripple in the passband, a steeper transition from the passband to the stopband can be achieved. The third criterion, which may be important, describes the ability of the filter to pass signals with a spectrum lying within the passband without distorting their shape due to phase shifts. You can also be interested in rise time, overshoot, and settling time.
Filter design methods are known that are suitable for optimizing any of these characteristics or combinations thereof. Truly smart filter selection doesn't happen as described above; As a rule, the required uniformity of the characteristic in the passband and the required attenuation at a certain frequency outside the passband and other parameters are first set. After this, 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 types of filters, namely the Butterworth filter (the flattest passband response), the Chebyshev filter (the steepest transition from passband to stopband), and the Bessel filter (the flattest lag time response). Any of these filter types can be implemented using various filter circuits; We will discuss some of them later. All of them are equally suitable for constructing low- and high-pass filters and bandpass filters.
Butterworth and Chebyshev filters. The Butterworth filter provides the flattest response in the passband, which comes at the cost of smoothness in the transition region, i.e. between passbands and delaybands. As will be shown later, it also has a poor phase-frequency response. Its amplitude-frequency characteristic is given by the following formula:
U out /U in = 1/ 1/2,
where n defines the filter order (number of poles). Increasing the number of poles makes it possible to flatten the portion of the characteristic in the passband and increase the steepness of the roll-off from the passband to the suppression band, as shown in Fig. 5.10.
Rice. 5.10 Normalized characteristics of Butterworth low-pass filters. Note the increase in the steepness of the characteristic rolloff with increasing filter order.
When choosing a Butterworth filter, we sacrifice everything else for the sake of the flattest characteristics. Its characteristic goes horizontally, starting from zero frequency, its inflection begins at the cutoff frequency ƒ s - this frequency usually corresponds to the -3 dB point.
In most applications, the most important consideration is that the passband ripple should not exceed a certain amount, say 1 dB. The Chebyshev filter meets this requirement, while some unevenness of the characteristic is allowed throughout the entire passband, but at the same time the sharpness of its break greatly increases. For the Chebyshev filter, the number of poles and the unevenness in the passband are specified. Allowing for increased unevenness in the passband, we obtain a sharper kink. The amplitude-frequency response of this filter is given by the following relation
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 unevenness of the characteristic in the passband. The Chebyshev filter, like the Butterworth filter, has phase-frequency characteristics that are far from ideal. In Fig. Figure 5.11 compares the characteristics of the 6-pole Chebyshev and Butterworth low-pass 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 very flat passband response 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 response as the frequency approaches ƒ c, and for the Chebyshev filter - ripples distributed over the entire passband). In addition, active filters built from elements whose ratings have some 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 in the Butterworth filter characteristic. In Fig. Figure 5.12 illustrates the effect 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 element parameters on the characteristics 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 greater steepness due to the fact that several equal-sized pulsations are distributed over the passband, the number of which increases with the order of the filter. Even with relatively small ripples (about 0.1 dB), the Chebyshev filter provides a much greater slope in the transition region than the Butterworth filter. To quantify this difference, assume that a filter is required with a passband flatness of no more than 0.1 dB and an attenuation of 20 dB at a frequency that differs by 25% from the cutoff frequency of the passband. Calculation shows that in this case a 19-pole Butterworth filter or just an 8-pole Chebyshev filter is required.
The idea that one can tolerate ripple in the passband for the sake of increasing the steepness of the transition section is taken to its logical conclusion in the idea of the so-called elliptic filter (or Cauer filter), in which ripple is allowed in both the passband and the delay in order to ensure the steepness of the transition section is even greater than that of the Chebyshev filter characteristic. With the help of a computer, elliptic filters can be designed as simply as the classical Chebyshev and Butterworth filters. In Fig. Figure 5.13 shows a graphical description of the amplitude-frequency response of the filter. In this case (low pass filter) the acceptable range of the filter gain (i.e. ripple) in the passband is determined, the minimum frequency at which the characteristic leaves the passband, maximum frequency, where the characteristic passes into the stopband, and the minimum attenuation in the stopband.
Rice. 5.13. Setting the filter frequency response parameters.
Bessel filters. As was established earlier, the amplitude-frequency characteristic of the filter does not indicate about it complete information. A filter with a flat amplitude-frequency response can have large phase shifts. As a result, the shape of the signal, the spectrum of which lies in the passband, will be distorted when passing through the filter. In situations where the waveform is of paramount importance, it is desirable to have a linear phase filter (constant delay time filter) available. Demanding a filter to ensure a linear change in the phase shift as a function of frequency is equivalent to requiring constant delay time for a signal whose spectrum is located in the passband, i.e., the absence of distortion of the signal shape. The Bessel filter (also called the Thomson filter) has the flattest part of the passband lag time curve, just as the Butterworth filter has the flattest frequency response. To understand the time domain improvement a Bessel filter provides, look at Fig. Figure 5.14 shows frequency-normalized lag time plots for 6-pole Bessel and Butterworth low-pass filters. The poor lag time characteristics of the Butterworth filter cause overshoot-type effects to occur when pulsed signals pass through the filter. On the other hand, you have to pay for the constancy of the delay times of the Bessel filter by 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 Bessel (1) and Butterworth (2) low-pass filters. The Bessel filter, due to its excellent time domain properties, produces the least waveform distortion.
There are many in various ways filter designs that attempt to improve the performance of a Bessel filter in the time domain, partially sacrificing constant delay time in order to reduce rise time and improve amplitude-frequency response. The Gaussian filter has almost as good phase characteristics as the Bessel filter, but with improved transient response. Another interesting class are filters that make it possible to achieve identical ripples in the delay time curve in the passband (similar to ripples in the amplitude-frequency characteristic of a 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 time is the use of all-pass filters, otherwise called time-domain equalizers. These filters have a constant amplitude-frequency response, and the phase shift can be changed according to specific requirements. Thus, they can be used to equalize the delay time of any filters, in particular Butterworth and Chebyshev filters.
Comparison of filters. Despite earlier comments about the transient response of Bessel filters, it still has very good time domain properties compared to Butterworth and Chebyshev filters. The Chebyshev filter itself, with its very suitable amplitude-frequency response, has the worst parameters in the time domain of all these three types of filters. The Butterworth filter makes a trade-off between frequencies and timing characteristics. In Fig. Figure 5.15 provides information on the performance characteristics of these three types of filters in the time domain, complementing the earlier graphs of amplitude-frequency characteristics. Based on these data, we can conclude that in cases where filter parameters in the time domain are important, it is advisable to use a Bessel filter.
Rice. 5.15. Transient comparison of 6-pole low-pass filters. The curves are normalized by reducing the attenuation value of 3 dB to a frequency of 1 Hz. 1 - Bessel filter; 2 - Butterworth filter; 3 - Chebyshev filter (ripple 0.5 dB).
