Scho also filter of butyworth, rozrakhunok that diagram. Course robot: Butterworth high frequency filter Formula for amplitude frequency response of Butterworth filter

For filters, you can fix the default parameters of the filter, the most important is the frequency response. We have already discussed at the statti how it will be possible to reduce the given filter to the low-pass filter prototype. The butt of the vimog to the amplitude-frequency characteristic of the low-pass filter prototype of the projected filter is aimed at the little 1.

Malunok 1. Butt of the normal amplitude frequency characteristics LPF

This graph shows the accumulation of the filter transmission rate up to the standard frequency ξ , de ξ = f / f v

On the hovering little 1 graph, you can see that the transmission rate of the smoothie can be set to the permissible unevenness of the transmission rate. Smoothies have a minimal effect of suppressing the signal, which will start. Real filter can be like a form. Smolder, you won’t overturn the boundaries of the assignments.

To reach a trivial hour of filtering was carried out by the method of selecting an amplitude-frequency characteristic for the addition of standard LANs (m-Lanka or k-Lanka). A similar method is called the aplication method. Win buv to finish folding and not giving optimal performance the quality of the crushed filter and the number of lanes. To that, the mathematical methods of approximation of the amplitude-frequency characteristic from the given characteristics have been broken down.

Approximation in mathematics is called the manifestation of folding fallowness as a function of its kind. The name of the function is simple. In the case of the filter design, it is important that the approximate function could easily be implemented circuitry. In addition, the function is realized with the additional zero and pole of the transmission coefficient of the four-pole, in the case of the filter. The stench is easy to implement behind the additional LC-contour or the ringing sounds.

A wider type of approximation of the filter's frequency response є approximation behind Butterworth. Some of the filters were called Butterworth Filter.

Filtree Butterworth

Due to the peculiarity of the amplitude-frequency characteristics of the Butterworth filter є the visibility of the minimum and maximum in the smoothie, the transmission and the shut-off. The decrease in the frequency response between the smog and the transmission of the number of filters to the doors is 3 dB. If the filter requires less than the value of the smoothness of the smooth flow, then the frequency of the filter is rotated f to vibrate at a given upper frequency of the smugy transmission. The frequency response approximation function for the LPF prototype of the Butterworth filter is as follows:

(1),

de ξ - The frequency is normal;
n- Filter order.

At the same real amplitude-frequency characteristic of the filter can be trimmed by multiplying the standard frequency ξ filter frequency. For the Butterworth filter of low frequencies, the frequency response approximation function is shown as follows:

(2).

Infectious beastly respect, when the filtrating is widespread, the understanding of the complex s-area is widely recognized, a circular frequency is put on the yak along the ordinate axis. jω, and on the abscis axis - the value, turned around by the quality factor. Thus, it is possible, by virtue of the main parameters of the LC-circuits, to enter before the warehouse of the filter circuit: adjustment frequency (resonant frequency) and quality factor. Go to the s-area, go for help.

A detailed map of the position of the Butterworth filter poles on the complex s-area is placed in. For us, it’s smarter than the poles of the whole filter are rostered on a single number at a time from one place to another. The number of poles starts with the order of the filter.

On little 2, the poles for the first order Butterworth filter have been rotated. The handrail is shown by the frequency response, which is related to the given rosetting of the poles on the complex s-area.

Malunok 2. Roztashuvannya poles and frequency response Butterworth filter first order

On little 2 it can be seen that for the filter of the first order, the pole is to blame for the adjustment to the zero frequency, and the quality factor is at fault for the single ones. The frequency response graph shows that the frequency of the adjustment of the pole is efficiently zero, and the quality of the pole is that, at the frequency of the standard Butterworth filter, that the transmission efficiency is high -3dB.

So the poles for the Butterworth filter are of a different order. Once the frequency of the alignment of the pole is vibrated on the cross-strand of a single pole with a straight line, then pass through the center of the pole at 45 °.

Malunok 3. Roztashuvannya poles and frequency response of the Butterworth filter in a different order

At the same resonance frequency of the pole, it is not close to the frequency of the normal filter. Won the door 0.707. The quality of the pole behind the graph of the rosetting of the poles at the roots is two times for the quality of the Butterworth filter pole of the first order, so the steepness of the drop in the amplitude-frequency characteristic is greater. (I respect the numbers at the right part of the graph. When the frequency is set to 2, it is adjusted to the level of 13 dB). The left part of the amplitude-frequency characteristics of the pole is flat. Tse z the poles injected, scho near the zones of negative frequencies.

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

Malunok 4. Roztashuvannya poles of Butterworth filter of the third order

You can see it from the graphs shown on the little ones 2 ... 5, with an increase in the order of the Butterworth filter, the steepness of the drop of the amplitude-frequency characteristic increases and the growth requires the quality of the lantern of a different order of the characteristic (contour), more The very growth of the required quality factor is interconnected with the maximum order of the filter, which can be realized. It is possible to realize the Butterworth filter up to the eighth - tenth order.

Filtri Chebisheva

In Chebishev filters, the approximation of the amplitude-frequency characteristic is carried out by the offensive rank:

(3),

With a wide amplitude-frequency characteristic of a real Chebishev filter, so it is possible to correct the Butterworth filter by multiplying the standard frequency ξ to the frequency of the filter, which can be broken up. For the Chebishev filter of low frequencies, the amplitude-frequency characteristic can be as follows:

(4).

Amplitude-frequency response of the Chebishev filter low frequencies characterized by a greater steep decline in the frequency range above the upper transmission frequency. Tsey vigrash reach for the rakhunok, show the uneven frequency response of the smooth flow. The inconsistency of the function of approximation of the frequency response of the Chebishev filter is to be determined by the higher quality of the poles.

A detailed diagram of the position of the poles of the approximate function of the Chebishev filter on the s-area is pointed at. For us, it is important that the poles of the Chebishev filter roztashovani on elipsi, there is a great deal of how to get out of all the standard frequencies. On the central axis, the elips pass through the frequency point and the lower frequency filter.

The standard version has a single point. Another way is to start with an uneven function of approximation of the frequency response of the smoothie throughput. Chim more is permissible unevenness in smoothies passing, tim mensha tsya. It is possible to "flatten" a single stake of Butterworth filter. The poles approach the frequency axis. The increase in the quality factor of the filter poles. Something more unevenness in smoothies throughput, tim more goodness of poles, more rapidity of growth in smoothness in smoothies, non-skipping filter of Chebishev. The number of poles in the AFC approximation function starts with the order of the Chebishev filter.

Slide means that Chebishev's first filter is dumb. The rotation of the poles and the frequency response of the Chebishev filter is in a different order. The stench reflects the maxima of the frequency response of the smoothie throughput. The filter has a different order, the frequency of the pole is ξ =0.707.

1 The order of the filter is significant. The order of the filter is the number of reactive elements of the LPF and HPF.

de
- the Butterworth function, which shows the permissible frequencies .

- Allowed extinguishing.

2 Drawn diagram of the filter of the trimmed order. With the practical implementation of the circuitry with a smaller number of inductances.

3 Rozrakhovuєmo permanent re-adaptation of the filter.

, mH

, nF

4 For ideal filter with 1 Ohm generator support, 1 Ohm support
a table of standard coefficients of the Butterworth filter has been added. In the skin row of the table, the efficiency is symmetrical, to the middle to increase, and then to change.

5 In order to know the elements of the diagram, it is necessary to constantly re-implement it by multiplying the table performance.

Filter order	Filter number m

Rosrahuvati parameters of the Butterworth low-frequency filter, where PP = 0.15 kHz, = 25 kHz, = 30 dB,
= 75 Ohm. Know
for three points.

29.3 Butterworth HPF.

Filtri HPF - central chotiripoles, for those in the range (
) there is little loss, and in the range (
) - great, so that the filter is guilty of letting it go to the navantazhennya struma high frequencies.

Since the high-frequency filter is guilty of passing high frequencies by the struma, then on the way to the struma, right in the middle of the installation, it is guilty of standing a frequency fallow element, which is good for passing high frequencies and disgusting low frequencies by the struma. This element is a capacitor.

F
HF T-sub

HPF P-podibny

The capacitor should be installed one after the other,
the same frequency
to change, it is easy to pass through the condenser with the high frequency streams. Put the inductance coil in parallel with the installation, splinters
and with increasing frequencies and increasing
In addition, the low-frequency streams are locked through the inductance and not consumed at the installation.

The design of the Butterworth LPF is similar to the design of the Butterworth LPF, it is carried out according to the same formulas, only

.

Rosrahuvati Butterworth HPF high-frequency filter, yaksho
Ohm,
kHz,
dB,
kHz. Know:
.

Busy Topic 30: Butterworth's Dark and Notch Filters.

Storinka 1 z 2

Apparently, the order of the filter goes from the necessary minds behind the graph for fading into smoothies in the book by G. Lem "Analog and digital filters" chapter 8.1 side 215.

Zrozumіlo, scho for the necessary extinguishing a filter of the 4th order is sufficient. The graph is displayed for the drop, if w s = 1 rad / s, and the frequency is indicated, for which it is necessary to be extinguished - 2 rad / s (apparently 4 and 8 kHz). Headline graph for transmission of Butterworth filter:

Visually schematic implementation of the filter:

active filter of low frequencies of the fourth order with a folding negative ringing ring:

The scheme is small for the amplitude-frequency characteristic, the elements, which can be entered before it, can be selected from no less high accuracy, but the plus of this scheme.

active filter of low frequencies of the fourth order with a positive ringing ring:

In the case of the scheme, the efficiency of the operative driver is strictly guilty of the mother, and the efficiency of the transmission of the scheme will not be more than 3. This scheme can be seen.

active low-frequency filter of the fourth order with an omnipresent negative ringing ring

Danny filter of motivations for chotiroh operatives, so that the foldability of the design of the scheme is improved, that is also visible.

With a vibrating filter with a foldable negative ringing ring.

Rozrahunok filter

The value of the transmission function

Write tabular values coefficients for the fourth order Butterworth filter:

a 1 = 1.8478 b 1 = 1

a 2 = 0.7654 b 2 = 1

(div. U. Titze, K. Schenk "Napivprovidnikova circuitry" table 13.6 side 195)

The main rotation of the transmission function of the low-pass filter of the fourth order:

(div. U. Titze, K. Schenk "Napivprovidnikova circuitry" table 13.2 side 190 and form 13.4 side 186).

The forward function of the first Lanka maє viglyad:

The forward function of another Lanka maє viglyad:

de w s is the circular frequency of the filter, w s = 2pf c.

Rozrakhunok nominals of details

Having adjusted the performance of the viraziv (2) and (3) the performance of the viraz (1), we can take it:

Features of transmission post-signal for cascades, їх add-ons A 0 maє dorіvnyuvati 10 per zavdannyam. The stench of negative, some of the cascades are invertible, the protest good is the positive transmission efficiency.

To decorate the circuit more beautifully, set the capacitors' capacities, while in order for the value of R 2 to be effective, you can

and apparently

Z chih of minds vibrate Z 1 = Z 3 = 1 nF, Z 2 = 10 nF, Z 4 = 33 nF.

Rozrahovuєmo meaning of supports for the first cascade:

Values ​​of supports in another cascade:

Vibir OU

When the op amp is vibrated, it is necessary to change the frequency range of the filter: the frequency of a single op-amp gain (for a single performance factor for a single unit) can be higher for an extra frequency and an increase in the filter efficiency.

Oskilki the maximum efficiency of the road is 3.33 and the frequency is 4 kHz, then we think better than all the available op amps.

Іnshim important parameter OU є yogo input opir. Vono can be more than ten times the maximum op_r of the circuit resistor.

The maximum op_r for the high-voltage circuit is 99.6 kOhm, and the input op_r of the op-amp is not less than 996 kOhm.

It is also necessary to provide the installation of the OS. For the current OU, the minimum opyr for the option is to become 2 coma. Vrahoyuchi, which opir R1 and R4 equal to 33.2 and 3.09 kOhm, the output stream of the operational pressure will be less than the maximum allowable.

According to the previous passport data (characteristics), OU K140UD601 was collected before the installed ones:

Up to. min = 50,000

R in = 1 MOhm

We will talk about the Butterworth filter, the order of the filters, decades and octaves, the third-order low-frequency filter with the layout and the circuit is shown in detail.

Entry

Near the outbuildings, like a vikoristovyut filters for form frequency spectrum the signal, for example, in systems of communication or control, the form or the width of the fall, is also called the "dark transition", for a simple filter of the first order, it can be used as a backward one or the necessary wide and active filters, broken with more than one. Tsi types of filters call out to yak filters "in the highest order" or "n-th order".

Filters order

Folding, or the type of filter, starts with the "order" of filters and stays in the form of a number of reactive components, such as a condenser or a coil of inductance of its design. We also know that the speed of the decay і, the same, the width of the smuga transition to lie in the serial number of the filter, and for a simple filter, the first order is in the standard speed of the decay 20 dB / decade or 6 dB / octave.

Todi for the filter, which is the n-th ordinal number, in case of matime there will be a drop rate of 20n dB / decade or 6n dB / octave. In this rank:

• first filter low decay rate 20 dB / decade (6 dB / octave)
• filter of a different order low decay rate 40 dB / decade (12 dB / octave)
• filter of the fourth order m roll-off frequency 80 dB / decade (24 dB / octave) thin.

Filters of the highest order, such as the third, quarters and n'yats, are inviting to form a path of cascade formation of single filters of the first and another order.

For example, two filters of lower frequencies of a different order can be cascaded to remove the filter of lower frequencies of the fourth order and so far. Unimportant to those who order the filter, which can be formed, not encircled, with an increase in the order, the size and parity will increase, and also the accuracy will decrease.

Decade and octave

Last comment about Decadesі Octaves... Over the frequency scale decade- A tenfold decrease (multiplied by 10) or a tenfold decrease (increased by 10). For example, from 2 to 20 Hz to become one decade, from 50 to 5000 Hz to become two decades (from 50 to 500 Hz, and then from 500 to 5000 Hz).

Octave- tse subnnya (multiply by 2) or decrease in vdvіchі (rose by 2) over the frequency scale. For example, from 10 to 20 Hz represents one octave, and from 2 to 16 Hz - three octaves (from 2 to 4, from 4 to 8 і, nareshty, from 8 to 16 Hz), the frequency is different. Be-like vipad, logarithmic The scale is widely used in the frequency area for the given value of the frequency of the robot with the help of filters and filters;

Oscillation resistors, for the first, second, third, or for the fourth order filter, for the fourth order of magnitude;

If the filters of the first and the other order, the filters of the upper frequencies of the third and fourth order are formed, we will simply exchange the position of the initial frequency of the components (resistors and capacitors) in the equivalent filter of the lower frequencies. Filters of the highest order can be designed, using procedures that were previously used in the manuals with a filter of low frequencies and filters of high frequencies. However, the out-of-the-box performance of the filters in the highest order є fiksovanim, Oscillations of all components that start with the frequency, however.

Filter approximations

We looked at the low-frequency and high-frequency filter circuits of the first order, and the resulting frequency and phase characteristics. An ideal filter giving bi us the specifics of the maximum smog letting through the area, the smallest smog letting through, as well as even more steep smog passing, so that the smog decay (swamp transition), and it is obvious that great number They are all right in the middle.

It is not surprising that in the linear design of analog filters there is a number of approximate functions, in which a mathematical approach is used for the best approach to the transmission function, as we need for the design of filters.

Such constructions vidomi yak Еліптичний, Butterworth, Chebishiv, Bessil, Kauer and a lot of them. Three of five "classic" functions of approximation of a linear analog filter only Butterworth filter and especially the design low frequency Butterworth filter there will be a look here as its function, most often it will be victorious.

Low-frequency Butterworth filter

Frequency response of the approximate function filter Butterworth It is also often called “maximally flat” (without pulsation) characteristic, the smog transmission is designed so that the frequency response is flat, but mathematically it can be from 0 Hz (DC) to a frequency of -3 dB without pulsations. More high frequencies beyond the boundaries of the point of increase, decrease to zero in smoothies at 20 dB / decade or 6 dB / octave. Moreover, there is a quality factor, Q is 0.707.

However, one of the main shortcomings of the Butterworth filter is those that are not within the reach of the wide area of ​​the smog, if the filter changes from the swamp to the swamp of the zupinka. There are also poor phase characteristics. Ideal frequency response, which is called the "celiac style" filter, is the standard Butterworth approximation for the lower orders of the filter hovering lower.

I respect that the order of the Butterworth filter, there are more cascade descents in the filter design, and the closer the filter goes to the ideal view of the "sense of style".

However, in practice, the ideal frequency response of Butterworth is inadequate;

De uzagalnene rіvnyannya, which represents the Butterworth filter "n-th" order, the frequency response is given yak:

De: n represents the order of the filter, ω is 2πƒ, and ε is the maximum throughput of the smog (A max).

If A max is assigned at a frequency, but for the most important points of the output is -3 dB (s), for the same ε as for the same unit, and also for ε 2 as for the same unit. However, if you now want the value of A max at the same value for the load power, for example, 1 dB or 1.1220 (1 dB = 20 * logA max), if the new value of ε is found behind the formula:

Pidstavlyayuchi danі to іvnyany, we will recognize:

Frequency response filter can be used mathematically transfer function with the standard of transmission of the sprung Function H (jω) and recorded by the viewer:

Note: (jω) can also be written as (s) for the value S-areas The resulting transfer function for the low frequency filter in a different order is set as:

Normalized Low Pass Butterworth Filter Polynomy

To help with the development of their low-frequency filters, Butterworth opened the standard tables of normalized low-frequency fields of a different order with respect to the value of the performance, as the display of the frequency / frequency response.

N	Normalized polynomies of the denominator at factorized forms
1	(1 + S)
2	(1 + 1.414 s + s 2)
3	(1 + s) (1 + s + s 2)
4	(1 + 0.765 s + s 2) (1 + 1.848 s + s 2)
5	(1 + s) (1 + 0.618 s + s 2) (1 + 1.618 s + s 2)
6	(1 + 0.518 s + s 2) (1 + 1.414 s + s 2) (1 + 1.932 s + s 2)
7	(1 + s) (1 + 0.445 s + s 2) (1 + 1.247 s + s 2) (1 + 1.802 s + s 2)
8	(1 + 0.390 s + s 2) (1 + 1.111 s + s 2) (1 + 1.663 s + s 2) (1 + 1.962 s + s 2)
9	(1 + s) (1 + 0.347 s + s 2) (1 + s + s 2) (1 + 1.532 s + s 2) (1 + 1.879 s + s 2)
10	(1 + 0.313 s + s 2) (1 + 0.908 s + s 2) (1 + 1.414 s + s 2) (1 + 1.782 s + s 2) (1 + 1.975 s + s 2)

Rozrahunok and low-frequency Butterworth filter circuit

Know the order of the active low-frequency Butterworth filter, whose aiming characteristics are: A max = 0.5 dB at a transmission frequency (ωp) 200 radian / s (31.8 Hz), and A min = -20 dB at a signal frequency (ωs) 800 radian / sec. Also design a schematic diagram of the Butterworth filter to display the image of the Vimogam.

Perceptible, the maximum smuga transmission is A max = 0.5 dB, as it is more efficient 1,0593 Remember that: 0.5 dB = 20 * log (A) at a frequency (ωp) of 200 rad / s, the value of epsilon ε is based on:

In a different way, the smog of the zupinka is minimal, A min = -20 dB, which is more efficient 10 (-20 dB = 20 * log (A)) at zupinka frequency (ωs) 800 rad / s or 127.3 Hz.

Substitution value y zagalne rivnyannya for the frequency response of the Butterworth filters, give us the following:

So, if n depends on the number, then we start to find the values ​​2.42 if n = 3 "Third order filter", the one for the stem filter Butterworth the third order, a filter of another order is required for a cascade of a filter step of the first order.

From the induced tables of normalized Butterworth polynomials of low frequencies, the third-order filter performance is given as (1 + s) (1 + s + s 2), and we can do it with 3-A = 1 or A = 2. B A = 1 + (Rf / R1), vibrating value for the resistor ringing bell Rf і resistor R1 gives us a value of 1 kOhm and 1 kOhm, apparently, yak: (1 kOhm / 1 kOhm) + 1 = 2.

We know that the cut-off frequency, the point -3 dB (ω o), can be found behind the additional formula 1 / CR, but we need to know ω o for the frequency of smog transmission ω p,

Thus, the frequency of the output of the cut is set at 284 rad / s or 45.2 Hz (284 / 2π), і, vikoristovuyu known formula 1 / RC, we can know the value of the resistors and capacitors for our circuit of the third order.

Beastly respect, scho the closest perevag up to 0.352 uF would be 0.36 uF or 360 nF.

I, nareshty, our low-frequency filter circuit Butterworth the third order with a cut-off frequency of 284 rad / s or 45.2 Hz, the maximum strength of the smog, the transmission of 0.5 dB, and the minimum strength of the smog of the zupinka of 20 dB, will be the offensive rank.

Thus, for our filter of low frequencies Butterworth of the 3rd order with a cut frequency of 45.2 Hz, C = 360 nF and R = 10 kOhm

Significant part of the theory of the development of digital BIX-filters (that is, the filter because of the endless impulse characteristic) is the need for intelligent methods in the development of filters without interruption. Tom in given given formulas will be introduced for some standard types of analog filters, including Butterworth, Bessel and Chebishev type I and II filters. Detailed analіz perevag that nedolіkіv sposobіv aproksimatsії assignments characteristics vіdpovіdnih CIM fіltram can know in ryadі robіt, prisvyachenih methods rozrahunku analog fіltrіv, will attempt to nizhche lishe short pererahovanі osnovnі vlastivostі fіltrіv skin type that navedenі rozrahunkovі spіvvіdnoshennya, neobhіdnі for otrimannya koefіtsієntіv analog fіltrіv.

Do not need to develop the standards of the low-frequency filter from the frequency of the sound, which is important Ω = 1 rad / s. As the function is approximated, as a rule, there is a square of the amplitude characteristic (for example, Bessel filter). We will take into account that the transfer function of the analog filter the rational change function S of the offensive type:

Low frequency Butterworth filter is characterized by the time that maximally smooth amplitude characteristic per cob of coordinates at the s-area. This means that all the same old characteristics from the amplitude characteristics of the cob of coordinates go to zero. The square of the amplitude characteristic of the standard (tobto we have a frequency of 1 rad / s) Butterworth filter door:

de n - Filter order. Analytically extending the function (14.2) to the entire S-area,

The poles (14.3) are located on a single number at the same time, one at one in one S-area ... Virazimo transfer function H (s) across poles S :

De (14.4)

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

a k 0 - The constant of the norm. Vikoristovuchi formulas (14.2) and (14.5), it is possible to formulate a chain of powers of Butterworth filters of lower frequencies.

The power of the lower frequency Butterworth filters:

1. Butterworth's filters may not have a pole (all zero transmitting functions are used for filtering at non-termination).

2. At a frequency Ω = 1 rad / s, the transmission efficiency of Butterworth filters is dorvnyuє (that is, at a frequency the amplitude is 3 dB).

3. Filter order n I will add the whole filter. For the sake of the order of the Butterworth filter, make sure to get rid of the loss of the singing weakening for the given frequency Ω t> 1. The order of the filter, so that the frequency is not guaranteed Ω = Ω t< уровень амплитудной характеристики, равный 1/А, можно найти из соотношения

Small. 14.1. Roztashuvannya poles of analog filter Butterworth of low frequencies.

Small. 14.2- Amplitude and phase characteristics, as well as the characteristic of the group response of the analog Butterworth filter of low frequencies.

Come on, for nothing, required at frequency Ω t = 2 rad / s to prevent weakening, which is expensive, A = 100. Todi

Rounded n at the great side to an integer number, we know, that weakening is not given to secure the Butterworth filter of the 7th order.

Decision... Vikoristovuchi yak rorakhunkovi characteristics 1 / A == 0.0005 (which will be attenuated by 66 dB) Ω t = 2, otrimaєmo n== 10.97. Rounding off yesє n = 11... In fig. 14.1 shows the rosetting of the poles of the rooted Butterworth filter s-area... Amplitude (on a logarithmic scale) and phase characteristics, as well as the characteristics of the group coverage of the filter are shown in Fig. 14.2.