Significant part of the theory of the development of digital BIX-filters impulse response) Vymagaє rozumіnnya methodіv rozrahunku filtrіv uninterrupted hour. Therefore, in this section, there will be developed formulas for several standard types of analog filters, including Butterworth, Bessel and Chebishev type I and II filters. A detailed analysis of the differences and shortcomings in the methods of approximating the given characteristics of the respective cim filters can be found in a series of studies associated with the methods of analysis of analog filters, so the main power of the filters of the skin type and the induction of the patterns will be less short. Unkovi spіvvіdnostnja, nebhіdnі for otrimannyа kofіtsієnіv analog іntіv.

Let it be necessary to adjust the normalization of the low-frequency filter from the frequency of sight, which is good Ω = 1 rad / s. As the function is approximated, as a rule, the square of the amplitude characteristic (excluding the Bessel filter) is used. Let's take into account that the transfer function of the analog filter is a rational change function S of the offensive form:

The Butterworth filters of the lower frequencies are characterized by the fact that they can achieve the most smooth amplitude characteristic on the cob of coordinates near the s-plane. Tse means that all significant changes in the amplitude characteristics of the cob of coordinates are equal to zero. The square of the amplitude characteristic of the normalized (so we can measure the frequency at a rate of 1 rad / s) Butterworth filter is good:

de n - Filter order. Analytically continuing the function (14.2) on the entire S-plane, we take

All poles (14.3) are on a single stake on the same width one in one in S-flats . Virazimo transfer function H(s) through the poles, which roam in the left nape S :

De (14.4)

De k = 1.2 ... n (14.5)

A k 0 - Normalization constant. Vikoristovuyuchi formulas (14.2) and (14.5), it is possible to formulate the number of powers of Butterworth filters of lower frequencies.

The power of Butterworth filters of low frequencies:

1. The Butterworth filters are more than poles (all zeros of the transfer functions of these filters are sorted on the inconsistency).

2. At a frequency of Ω = 1 rad / s, the Butterworth filter transmission coefficient is better (that is, at a frequency over time, the amplitude characteristic drops by 3 dB).

3. Filter order n override the entire filter. The correct order of the Butterworth filter is to ensure the safety of the attenuation at the current specified frequency Ω t > 1. The order of the filter, which ensures the safety of the frequency Ω= Ω t< уровень амплитудной характеристики, равный 1/А, можно найти из соотношения

Mal. 14.1. Rotation of the poles of the analog filter Butterworth of lower frequencies.

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

Come on, for example, required at frequency Ω t = 2 rad/s take care of the weakened, which is healthy A \u003d 100. Todi

Rounded up n y the great side up to an integer number, we know that the task is weakened to ensure the Butterworth filter of the 7th order.

Solution. Vicorist characteristics 1/A == 0.0005 (which shows a weakening of 66 dB) and Ω t = 2, taken n== 10.97. Rounded yes n = 11. On fig. 14.1 shows the expansion of the poles of the expanded Butterworth filter s-flats. Amplitude (on a logarithmic scale) and phase characteristics, as well as the characteristic of the group fading of the filter, are presented in Fig. 14.2.

The frequency response of the Butterworth filter is described by equals

Features of the Butterworth filter: non-linear phase response; frequency zrіzu not lie in the kіlkostі polesіv; The colicky character of the transient characteristic with a stepped input signal is increased to the greater order of the filter.

Chebishev filter

The frequency response of the Chebishev filter is described by equals

,

de T n 2 (ω/ω n ) – Chebishev polynomial n th order.

The Chebishev polynomial is calculated using the recursive formula

Peculiarities of the Chebishev filter: - increased the unevenness of the PFC; hvilepodіbna characteristic of smuga transmission. What is more the coefficient of uneven frequency response of the filter in smooth transmission, moreover, a sharp decline in the transitional region with the same order. Collation of the transitional process with a stepwise input signal is stronger, lower Butterworth filter. Goodness of the poles of the Chebishev's filter, lower of the Butterworth filter.

Bessel filter

The frequency response of the Bessel filter is described by equals

,

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

The Bessel polynomial is calculated using the recursive formula

Features of the Bessel filter: to achieve equal AFC and PFC, which are approximated by the Gaussian function; phase sound filter proportional to frequency, tobto. The filter can be a frequency-independent group hour of the fade. The frequency is changed by changing the number of filter poles. The drop in the frequency response of the filter sounds more gentle, lower for Butterworth and Chebishev. Particularly good filter is suitable for pulsed lasers and phase-sensitive signal processing.

Cauer filter (elliptical filter)

Cool look of the Cauer filter function

.

Features of the Cauer filter: uneven frequency response in smooth transmission and smooth trimming; the largest drop in frequency response due to the use of filters; implement the necessary transfer functions with a smaller order of the filter, lower interval of the filter of other types.

Designated filter order

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

.

Order of the Chebishev filter

.

For the Bessel filter, there is no formula for the breakdown of the order, it is necessary to create a table of performance in the order of the filter by the minimum required at a given frequency, at the time of the dimming, as a single value and the level of losses in dB).

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

Permissible interval for the group time of the fade at the given frequency ω ω cp h ;

There may be tasks equal to the attenuation of the filter transfer coefficient y dB at the frequency ω , normalized ω cp h .

From these data, the necessary order of the Bessel filter is determined.

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

On fig. 12.4, 12.5 a typical circuit of LPF cascades is shown.

A) b)

Mal. 12.4. Butterworth, Chebishev and Bessel low-pass cascades: A - 1st order; b - 2nd order

A) b)

Mal. 12.5. Kauera low-pass cascade: A - 1st order; b - 2nd order

A glaring view of the transfer functions of the LPF of Butterworth, Chebishev and Bessel of the 1st and 2nd order

,
.

Highlighted view of the transmission functions of the Cauer low-pass filter of the 1st and 2nd order

,
.

The key feature of the Cauer filter of the 2nd order is the filter, which is fenced, є those transfer function Cauer filter frequency change Ω s ≠ 1.

Butterworth, Chebishev and Bessel rozrahunka LLF technique

This technique was developed on the basis of the coefficients indicated in the tables and is valid for Butterworth, Chebishev and Bessel filters. The method of rozrahunku filters Cauer is induced okremo. Rozrahunok LLF Butterworth, Chebishev and Bessel start from the appointed order. For all filters, the parameters of the minimum and maximum attenuation of that frequency are set. For Chebishev filters, the coefficient of AFC unevenness in smooth transmission is additionally indicated, and for Bessel filters, the group fade hour. Next, the transfer function of the filter is determined, as it can be taken from the table, and the first and second order cascades are expanded, the next order of expansion is completed:

Depending on the order and type of the filter, the schemes of the first cascade are selected, with which filter of the paired order is added n/2 cascades of the 2nd order, and the unpaired order filter - from one cascade of the 1st order i ( n– 1) / 2 cascades of the 2nd order;

For the cascade of the cascade of the 1st order:

After the selected type and order of the filter, the value is displayed b 1 cascade of the 1st order;

Changing the area, the denomination of the capacity is selected C that razrakhovuetsya R for the formula (you can choose i R, but it is recommended to choose C, with accuracy measurement)

;

Strength coefficient is calculated Before at U 1 cascade of the 1st order

,

de Before at U- Coefficient of filter strength in general; Before at U 2 , …, Before at Un- Coefficients of strengthening cascades of the 2nd order;

For the implementation of the settlement Before at U 1 it is necessary to set the resistors, depending on the offensive

R B = R A ּ (Before at U1 –1) .

For the cascade of the cascade of the 2nd order:

Changing the area that I borrow, the denominations of capacities are selected C 1 = C 2 = C;

Viberyutsya for the tables of coefficients b 1 iі Q pi for cascades of the 2nd order;

For a given rating of capacitors C break down resistors R behind the formula

;

For the selected type of filter, it is necessary to set the power factor Before at Ui = 3 – (1/Q pi) skin cascade of the 2nd order

R B = R A ּ (Before at Ui –1) ;

For Bessel filters, it is necessary to multiply the denominations of all capacities by the required hour of trimming.

At the filters of the rozrahunok, start the sound from the settings of the parameters of the filter, the most important is the frequency response. As we have already discussed at the article, the order is reduced from the given filter to the LPF prototype. The butt could up to the amplitude-frequency characteristic of the low-pass filter prototype of the design filter was pointed at little 1.

Figure 1. An example of a normalized amplitude-frequency characteristic of a low-pass filter

On this graph, the delay of the filter transmission coefficient to the normalized frequency ξ , de ξ = f/f V

On the hovered little graph 1, it can be seen that the smoothness of the bandwidth is set to allow uneven transmission coefficient. The smoothness of the blockage is set to the minimum coefficient of suppression of the signal that matters. A real filter may be a mother-like form. Golovnya, so that she didn’t overthink between tasks, she could.

To finish the last three hours of filtering was carried out by the method of selection of the amplitude-frequency characteristics for the help of standard lankas (m-lanka or k-lanka). A similar method was called the application method. Vіn buv dosit folded and not giving the optimal spіvvіdnoshnennia of the quality of the broken filter and the number of lanoks. Therefore, mathematical methods and approximation of the amplitude-frequency characteristic from the given characteristics were developed.

Approximation in mathematics is called the phenomenon of collapsible fallowness as a familiar function. Sound this function is simple. In the case of filter expansion, it is important that the approximating function could easily be implemented in circuitry. Therefore, the functions are implemented with additional zeros and poles of the four-terminal transmission coefficient, at the same time as the filter. The stench is easily realized with the help of LC-contours or with return links.

The widest type of approximation of the frequency response of the filter is the Butterworth approximation. Similar filters took away the name Butterworth filter.

Butterworth filters

The characteristic feature of the amplitude-frequency characteristic of the Butterworth filter is the presence of minimums and maxima in the smoothing of the transmission and trimming. The decrease in the frequency response between smugs is 3 dB. As the filter type requires a smaller value of the unevenness of the transmission smoothness, then turning the filter frequency f be selected higher than the specified upper frequency of the smog of transmission. The function of approximating the frequency response for the LPF prototype of the Butterworth filter looks like this:

(1),

de ξ - Normalized frequency;
n- Filter order.

With which real amplitude-frequency characteristic of the filter can be taken away by multiplying the normalized frequency ξ frequency of the filter. For the Butterworth filter of lower frequencies, the frequency response approximation function looks like this:

(2).

At the same time, it is brutally respectful that when the filters are expanded, the understanding of the complex s-plane is widely understood, on which the circular frequency is added along the y-axis. jω, and on the abscissa axis - the value wrapped by the quality factor. In this way, you can assign the main parameters of the LC circuits, which are included in the stock of the filter circuit: the frequency of the adjustment (resonant frequency) and the quality factor. Crossing at the s-flat is required for help.

Detailed view of the position of the poles of the Butterworth filter on the complex s-plane is drawn in . For us, it’s a smut that the poles of the filter are roztashovanі on a single stake on a equal vіdstanі one in one. The number of poles is determined by the order of the filter.

On little 2, the polarization of the poles for the Butterworth filter of the first order was introduced. The order is shown by the frequency response, which confirms the given distribution of the poles on the complex s-plane.

Malyunok 2. Rotation of the pole and frequency response of the Butterworth filter of the first order

On the little 2 you can see that for the filter of the first order, the pole is to blame but the adjustments to the zero frequency and that the quality factor is to be increased to one. On the graph of the frequency response, it can be seen that the frequency of the adjusted pole is effectively equal to zero, and the quality factor of the pole is such that at the frequency in front of the normalized Butterworth filter, it is equal to one, the transmission coefficient is equal to −3dB.

So the poles for the Butterworth filter are in a different order. The second time the frequency of the pole alignment is selected on the crossbar of a single stake from a straight line, which passes through the center of the stake under the edge of 45°.

Malyunok 3. Rotation of the poles and frequency response of the Butterworth filter in a different order

At times, the resonant frequency of the pole is spread near the frequency of the normalized filter. Won cost 0.707. The quality factor of the pole behind the graph of the expansion of the poles at the roots is twice as high as the quality factor of the pole of the Butterworth filter of the first order, so the steepness of the decline in the amplitude-frequency characteristic is greater. (Respect for the numbers on the right side of the graph. When the frequency is higher than 2, the higher is 13 dB) The left part of the amplitude-frequency characteristic of the pole looks flat. Tse with a splash of poles, but in the zone of negative frequencies.

The rotation of the poles and the amplitude-frequency characteristic of the Butterworth filter of the third order are shown in figure 4.

Malyunok 4. Rotation of the poles of the Butterworth filter of the third order

As can be seen from the graphs shown on the little ones 2 ... 5, with an increase in the order of the Butterworth filter, the steepness of the decay of the amplitude-frequency characteristic increases and the Q-factor of the lancet increases in a different order (contour), which implements the pole of the transmission characteristic of the filter. The most important Q-factors require the maximum order of the filter, which can be implemented. Ninі vdaєtsya to implement Butterworth filters up to the eighth - tenth order.

Filter Chebisheva

For Chebishev filters, the approximation of the amplitude-frequency characteristic is carried out in the following order:

(3),

With any amplitude-frequency characteristic of a real Chebishev filter, the same as in a Butterworth filter can be taken by multiplying the normalized frequency ξ on the frequency of the filter, which is being expanded. For the Chebishev filter of low frequencies, the amplitude-frequency characteristic can be calculated as follows:

(4).

The amplitude-frequency characteristic of the low-pass Chebishev filter is characterized by a steeper decline in the frequency range above the upper pass frequency. Tsey wins reach for the rahunok the appearance of uneven frequency response in smooth transmission. The unevenness of the function of approximation of the frequency response of the Chebishev filter is due to the greater quality factor of the poles.

A detailed plot of the position of the poles in the approximating function of the Chebishev filter on the s-plane is shown in . For us, it is important that the poles of the Chebishev filter are placed on the ellipse; On the ciy axis elips pass through the frequency point in front of the lower frequency filter.

For the normalized variant, the point is the same. Another thing is the unevenness of the function of approximation of the AFC in the smooth transmission. The more errіvnomіrnіst is permissible in smuzі transmission, the less everything. It seems that "flattening" of a single stake of the Butterworth filter. The poles are approaching up to the frequency axis. Tse v_dpovidaє increase in the quality factor of the poles of the filter. The greater the unevenness of the smoothness of the transmission, the greater the quality factor of the poles, the greater the speed of the increase in the rate of extinction of the Chebishev filter in the smoothie. The number of poles in the AFC approximation function is determined by the order of the Chebishev filter.

The next step is to indicate that there is no Chebishev filter of the first order. Raztashuvannya poles and the frequency response of the Chebishev filter of a different order brought a small 5. The characteristic of the Chebishev filter is tsikavat tim, which clearly shows the frequency of the poles. The stinks give the peaks of the frequency response of the smooze transmission. In a filter of a different order, the frequency of the pole is different ξ =0.707.

Butterworth filter

Transfer function of the Butterworth low-pass filter n th order is characterized by virase:

The amplitude-frequency characteristic of the Butterworth filter can be so powerful:

1) For whatever order n frequency response

2) at the frequency zrіzu u = u s

The frequency response of the LPF changes monotonously with increasing frequencies. Therefore, Butterworth filters are called filters with the most flat characteristics. Small 3 shows the graphs of the amplitude-frequency characteristics of the LPF Butterworth 1-5 orders. Obviously, the higher the order of the filter, the more accurately the frequency response of the ideal low-pass filter is approximated.

Malyunok 3 - AFC for the Butterworth filter of low frequencies in 1 to 5

Small 4 shows a schematic implementation of the HPF Butterworth.

Malyunok 4 - HPF-II Butterworth

The advantage of the Butterworth filter is the smoothest possible frequency response at smog frequencies, and the reduction is practically to zero at smothering smothering frequencies. The Butterworth filter is the only one of filters, which takes the form of the frequency response for larger high orders (because of the greater steep decline in the characteristics of the damped smoothness) as well as a rich variety of other types of filters (Bessel filter, Chebishev filter, elliptical filter) can vary the shape of the frequency response at different orders.

However, in comparison with Chebishev I and II types of filters or an elliptical filter, the Butterworth filter may have a greater gentle decline in characteristics and this is due to the greater order (which is more complicated in implementation) in order to ensure required characteristics at the frequencies of the smuga, it is suppressed.

Chebishev filter

The square of the module of the transfer function of the Chebishev filter is determined by the frequency:

de is the Chebishev polynomial. The module of the transfer function of the Chebishev filter is equal to one at low frequencies, de-transformed to zero.

Chebishev's filters sound victorious there, where it is necessary for an additional filter of a small order to provide the necessary characteristics of the frequency response, zocrema, good attenuation of the frequencies of the swirl smothering, and with it, the smoothness of the frequency response at the frequencies of the swag of the transmission and smothering is not so important.

The filters of Chebishev I and II are disassembled.

Chebishev's filter of the 1st kind. The most common is the modification of Chebishev's filters. In a smoother, the transmission of such a filter shows pulsations, the amplitude of which is indicated by the pulsation indicator. In the case of the analog electronic Chebishev filter, the order is higher than the number of reactive components, which are different during its implementation. A larger steep decline in the characteristic can be subtracted to allow pulsations not only in the transmission smoothness, but also in the smothering smoothness, adding zeros to the transmission filter function on the apparent axis jsh in the complex plane. Tse, however, reduced to a lesser effective suppression of smug suffocation. Removing the filter is an elliptical filter, also known as the Cauer filter.

The frequency response for the Chebishev filter of lower frequencies of the first kind of the fourth order is represented by small 5.

Figure 5 - AFC for the Chebishev filter of lower frequencies of the 1st kind of the fourth order

The Chebishev filter of the ІІ genus (inverse Chebishev filter) is faster than the lower Chebishev filter of the І genus through a smaller steep decline in the amplitude characteristic, which leads to an increase in the number of components. In the new daytime pulsation, the smoothness has a transmission, the proteus has a strangulation in the smooth.

The frequency response for the Chebishev filter of low frequencies of the second kind of the fourth order is represented by small 6.

Malyunok 6 - frequency response for the Chebishev filter of lower frequencies of the second kind

On a small scale 7 representations of the circuit implementation of the Chebishev HPF of the 1st and 2nd order.

Malyunok 7 - Chebishev HPF: a) I order; b) II order

The power of the frequency characteristics of the Chebishev filters:

1) The smoothness of the frequency response may have a balanced character. On the interval (-1? u? 1) є n the point at which the function reaches the maximum value, which is equal to 1, or the minimum value, which is equal. If n is unpaired, if n is paired;

2) the value of the frequency response of the Chebishev filter at a frequency of higher

3) When the function is monotonously changing and the value of zero.

4) The parameter e indicates the unevenness of the frequency response of the Chebishev filter in the transmission smear:

The equalization of the frequency response of the Butterworth and Chebishev filters shows that the Chebishev filter provides more attenuation in the smoothness of the transmission, the lower Butterworth filter of this order. The mismatch of Chebishev's filters is due to the fact that their phase-frequency characteristics of the transmission smear are significantly different from those of the linear ones.

For Butterworth and Chebishev filters, there are report tables, in which the coordinates of the poles and the coefficients of the transfer functions of different orders are indicated.

Institute of Color Metals and Gold SibFU

Department of Automation of Manufacturing Processes

Types of filters  LPF Butterworth  LPF Chebisheva I type  Minimum filter order  LPF from MOS 

 LPF on INUN  Biquadratic LPF  Adjustment of filters 2nd order  unpaired LPF 

 LPF Chebisheva II type  Eliptic LPF  Eliptic LPF on INUN  Elіptichnі low-pass filter on 3 capacitors  Biquadratic elliptical LPF  Nalashtuvannya LLF Chebishev II type and elliptic 

 Adjustment of filters 2nd order  All-pass filters  LPF modeling  Creation of schemes 

 Rozrahunok transitional cold  Rozrahunok frequency x-to  Vikonanny roboti  Control nutrition 

Laboratory robot №1

”Signal filtering at the middle of the Micro-Cap 6/7”

Meta roboti

1. Check the main types and characteristics of filters

2. Continue modeling filters in the Micro-Cap 6 core.

3. Follow up the characteristics of active filters in the medium Micro-Cap 6

Theoretical performance

1. Typical characteristics of filters

Signal filtering plays an important role in digital systems management. The stench of the filter vikoristovuyutsya usunennya vipadkovy pardons vimiru (supervision of signals of transition, noise) (Fig. 1.1). Separate hardware (circuitry) and digital (software) filtering. In the first type, the vicorist type has electronic filters with passive and active elements, in the other type, different software methods and changes are installed. Hardware filtering is installed in the USO modules (attachment of communication with the object) of controllers and subdivisions of data collection systems and control.

Digital filtering victorious in UVM upper level APCS. Whose robots are reportedly looking at the power of the hardware filtering.

The following types of filters are available:

low pass filters - LPF (pass low frequencies and trim high frequencies);

filters upper frequencies(pass high frequencies and trim low frequencies);

band-pass filters (pass a range of frequencies and cut off frequencies, spreading more and lower than the range of smuga);

band-blocking filters (like blocking a range of frequencies and passing frequencies, spreading more and lower than the range of smuga).

The transfer function (TF) of the filter may look like:

de ½ H(j w)½- module PF chi frequency response; j (w) - PFC; w - peak frequency (rad / s), tied to the frequency f (Hz) spacing w = 2p f.

P F of the implemented filter can be seen

de Aі b - constant values, and T , n = 1, 2, 3 ... (m £ n).

Steps of the polynomial of the banner n sets the order of the filter. Chim vіn vishchy, tim more beautiful frequency response, but folded scheme, and varst vishcha.

The range of several frequencies, for which signals pass, - the range of transmissions for which frequency response values ​​\u200b\u200b½ H(j w)½ is large, but in an ideal situation it is constant. The frequency ranges, for which signals are smothered, - if they are smoothed out, their frequency response value is small, and ideally it is equal to zero.

Frequency response of real filters against theoretical frequency response. For LPF, the ideal and real frequency response is shown in fig. 1.6.

For real filters, smuga has a bandwidth - the frequency range (0 -  s), the value of the frequency response is greater for a given value A 1 . Smuga zatrimuvannya - the same frequency range ( 1 -∞), in which frequency response is less than - A 2 . The frequency interval for the transition from the smog of passing to the smog of zatrimannya ( c - 1) is called the transition region.

The best way to characterize the filters is to change the amplitude of the vicarious gassing. Exhaust in decibels (dB) is assigned to the formula

Amplitude value A = 1 a= 0. As A 1 = A/
= 1/= 0.707, then extinguishing at frequency w c:

The ideal and real characteristics of the low-pass filter with different extinguishing stages are shown in fig. 1.7.

Mal. 1.8. LPF ( A) and yoga frequency response ( b)

Passive filters (Fig. 1.8, 1.9) are created on the basis of passive R, L, C elements.

At low frequencies (below 0.5 MHz), the parameters of the inductance coils are not satisfactory: large differences and variations in the characteristics of ideal ones. Coils of inductance are badly attached to the integrated coil. The simplest low-pass filter (LPF) and its frequency response are shown in fig. 1.8.

Active filters are created on the base R, C elements and active elements - operational subsidiaries (OS). OU due to mother: high coefficient of strength (in 50 times more, lower in the filter); high speed increase in output voltage (up to 100-1000 V/µs).

Mal. 1.9. T- and P-like LPF

Active low-pass filters of the first and other orders are shown in fig. 1.10 - 1.11. Pobudova filter n-th order cascading connections Lanok N 1 , N 2 , ... , N m H 1 (s), H 2 (s), ..., N m ( s).

Double order filter P > 2 revenge n/ 2 lanoks of a different order, cascaded. Unpaired order filter P > 2 revenge ( P - 1) / 2 lanks of another order, that one lanka of the first order.

For filters of the first order PF

de Atі Z - constant numbers; P(s) is a polynomial of another or smaller degree.

The low-pass filter has the maximum suppression of the smoothing bandwidth a 1 does not exceed 3 dB, but fades out at the smoother a 2 to be in the range of 20 to 100 dB. LPF gain coefficient for the value of the yogo transfer function at s = 0 chi value of yoga frequency response at w = 0 , then . dorivnyuє A.

The following types of LPF are distinguished:

Butterworth- moyut monotonous frequency response (Fig. 1.12);

Chebisheva (type I) - frequency response to compensate for pulsations in smooth transmission, and monotonous in smooth trimming (Fig. 1.13);

inverse of Chebishev(type II) - the frequency response is monotonous in the transmission fluid and may pulsate in the trimming fluid (Fig. 1.14);

elliptical - Frequency response may ripple as a smooth transmission, and a smooth trim (Fig. 1.15).

Butterworth LF filter n-th order can frequency response of this kind

PF of the Butterworth filter as a polynomial filter is more advanced

For n = 3, 5, 7 PF normalized Butterworth filter is more expensive

de parameters e ta TO - constant numbers, and Z P- Chebishev polynomial of the first class P, equal

Rozmax R You can change it by selecting the value of the parameter to finish small.

The minimum allowable extinction in a smooth transmission - a constant range of pulsations - is shown in decibels

.

The PFs of the Chebishev and Butterworth LF filters are identical in form and are described by virases (1.15) - (1.16). The frequency response of the Chebishev filter is better than the frequency response of the Butterworth filter of the same order, because the first one has the width of the transition region. However, the Chebishev filter has a higher (non-linear) PFC than the Butterworth filter.

The AFC of the Chebishev filter of this order is better than the AFC of Butterworth, the chips of the Chebishev filter have the same width of the transition region. However, the PFC of the Chebishev filter is higher (non-linear) in the same way as the PFC of the Butterworth filter.

PFC of the Chebishev filter for 2-7 orders of guidance in Fig. 1.18. For por_vnyannya in fig. 1.18 the dashed line shows the PFC of the Butterworth filter of the sixth order. You can also indicate that the PFC of the Chebishev high order filters is higher than the PFC of the low order filters. It should be noted that the frequency response of the high-order Chebishev filter is better than the frequency response of the low-order filter.

1.1. SELECT MINIMUM FILTER ORDER

Based on fig. 1.8 and 1.9 it is possible to create a non-trivial visnovka, which has a higher order of Butterworth and Chebishev filters, shorter than their frequency response. However, the greater order complicates the circuit implementation and subsequently promotes variability. In this rank, it is important to choose the minimum necessary filter order, which we are pleased with the task of the helpers.

Go to the images in fig. 1.2 global characteristics set the maximum allowable bandwidth for the sms a 1 (dB), minimum allowable fade in smooth trim a 2 (dB), frequency zrіzu w z (rad / s) or f c (Hz) is the maximum allowable width of the transition region T W

de logarithms can be either natural or decimal.

Equation (1.24) can be written as

w w / w 1 = ( T w/w c) + 1

that otrimane spіvvіdnoshnja pіdstaviti (1.25) for the knowledge of the fallow order P in the width of the transition region, but not in the frequency w 1 . Parameter T W / w w is called normalized the width of the transitional region is a dimensionless value. Otzhe, T W and w can be set in radians per second, і in hertz.

Similar rank based on (1.18) for K = 1 know the minimum order of the Chebishev filter

and from (1.25) next, that the Butterworth filter, which satisfies them, is the mother of the offensive minimum order:

I know the nearest number, otrimuemo P= 4.

This stock clearly illustrates the superiority of the Chebishev filter over the Butterworth filter, as the main parameter is the frequency response. The Chebishev filter ensures the same coolness of the transmission function as the Butterworth filter has a dual folding.

1.2. LPF WITH GREAT RINGER

І WITHOUT ANY FACILITY COEFFICIENT

Mal. 1.11. LPF from MOS in a different order

There are a lot of ways to encourage active low-frequency filters of Butterworth and Chebishev. Further, the deeds of the most stagnant nine will be reviewed wild schemes, Starting from simple (from the point of view of the number of necessary circuit elements) and moving to the most folding.

For filters of a higher order, equalization (1.29) describes the PF of a typical lanka of a different order, de Before - yoga strength coefficient; Atі WITH - coefficients of lanka, inferred from the advanced literature. One of the largest simple circuits active filters, which implement the PF of lower frequencies zgіdno (1.29), is shown in fig. 1.11.

Tsya scheme realizuє rivnyannia (1.29) s inverting strength coefficient - Before(Before> 0) that

Opіr, scho you are happy with (1.30), dorivnyuє

Dotsilny pіdhіd u to set the nominal value of the capacity C 2 close to 10/ f c uF and choose the largest actual nominal value of the capacitance C 1 that satisfies you (1.31). Opir may be close to the value calculated for (1.31). The higher the order of the filter, the more critical the higher the order of the filter. As for the actuality of the daily calculation of the nominal value of the supports, it is necessary to indicate that the value of the supports can be multiplied by the total coefficient for the mind, that the values ​​of the capacities are divided by that very coefficient.

As a butt, it is acceptable that it is necessary to expand the Chebishev filter with an MOS of a different order with an uneven transmission of 0.5 dB, a light transmission of 1000 Hz, and a gain coefficient equal to 2. Before\u003d 2, w z \u003d 2π (1000), but also for addition A, it is known that Y \u003d 1.425625 and Z \u003d 1.516203. Variable nominal value C 2 = 10/f c\u003d 10/1000 \u003d 0.01 uF \u003d 10 -8 F, s (1.32) is acceptable

Now it is acceptable that it is necessary to expand the Butterworth filter of the sixth order with MOS, with a frequency of 3 f c= 1000 Hz K= 8. Vіn folded from three lanoks of a different order, skin from PF, so they are equal (2.1). Vibero coefficient of skin strength K= 2, which ensures the necessary coefficient of strength of the filter itself 2∙2∙2=8. Z supplement A for Persian Lanka is known At= 0.517638 ta Z = 1. Newly select the nominal value of the capacity Z 2 \u003d 0.01 μF і in the second direction s (2.21) we know Z 1 \u003d 0.00022 microfarads. Set the nominal value of the capacity Z 1 \u003d 200 pF and (2.20) we know the value of the supports R 2 = 139.4 kΩ; R 1 = 69.7 kΩ; R 3 = 90.9 com. Two other lanes are opened in a similar way, and then the lanes are cascaded to implement the Butterworth filter of the sixth order.

Due to its apparent simplicity, the MOS filter is one of the most popular types of filters with an inverted power factor. Vіn mає takozh pevnі perevagi, and the stability of the characteristics and low vihіdny povniy opіr is good; in this order, it can be combined cascaded with other links for the implementation of a higher order filter. The shortcoming of the scheme is that it is impossible to reach a high value of the quality factor Q without a significant increase in the value of the elements of that high sensitivity to the current change. To achieve good results, the coefficient of strength Before

Skorigovana LPF-filter. ... ISO-structure, є possibility of regulating the strength of that filter when changing nominal values minimal ... filter on microcircuits type... may be the same order quantities that i... filtersChebishevaі Butterworth, ...