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 irrationality of the PFC; hvilepodіbna characteristic of smuga transmission. The higher the coefficient of uneven frequency response of the filter in smooth transmission, the sharper 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 (elliptic filter)

Cool look of the Cauer filter function

.

Features of the Cauer filter: uneven frequency response in smooth transmission and smooth trimming; the highest drop in frequency response due to the use of filters; implement the necessary transfer functions for the lower order of the filter, lower for the lower order of filters of the 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)

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

a) b)

Rice. 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 role of the Cauer filter of the 2nd order in the filter, which is obstructed, is those that are in the transmission function of the Cauer filter to change frequencies Ω s ≠ 1.

Butterworth, Chebishev and Bessel rozrahunka LLF technique

This technique was suggested on the basis of coefficients, which were inferred from the tables, and it 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. Dali vyznazhetsya transfer function filter, as can be taken from the table, and the cascades of the 1st and 2nd order are opened, the next order of the order is completed:

In the fallow order, the type of filter is selected for the schemes of its cascades; n/2 cascades of the 2nd order, and the unpaired order filter - from one cascade of the 1st order i ( n– 1) / 2 cascades in 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 and be insured 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, the nominal values ​​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 nominal values ​​of all capacities by the group hour of trim.

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 greater the order of the filter, it is more accurate to approximate the frequency response of an ideal low-pass filter.

Figure 3 - Frequency response for the Butterworth filter of low frequencies in the order of vіd 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 maximally smooth frequency response at the frequencies of smog transmission and її reduction to zero at the frequencies of smog smothering. The Butterworth filter is the only one of filters, which takes the shape of the frequency response for larger high orders (because of the greater steep decline in the characteristics on the smothering smear) as well as other filter varieties (Bessel filter, Chebishev order filter, e.A.Ch.

However, in combination 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 a larger order (which is more complicated in implementation) in order to ensure the required characteristics at smothering frequencies.

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, which turns to zero.

Chebishev's filters sound vikoristovuyutsya there, it is necessary for an additional filter of a small order to provide the necessary characteristics of the frequency response, zocrem, garne of frequency suppression for smug suffocation, and with it, the smoothness of the frequency response at the frequencies of smug transmission 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 the case of the transmission of such a filter, ripples are visible, the amplitude of which is indicated by the indicator of pulsations e. In the case of an analog electronic Chebishev filter, the order is higher than the number of reactive components, vicorisings during 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 transfer function on the apparent axis jsh in the complex plane. Tse, however, reduced to a lesser effective strangulation smog suffocation. The removal of 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 - Frequency response for the Chebishev filter of lower frequencies of the first 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.

Figure 6 - AFC for the Chebishev filter of low 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) second 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 changes monotonously 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 the same 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 systems are inferred.

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
- Butterworth function, which determines the allowable frequency .

- Let's go out.

2 We draw the scheme of the filter in the selected order. With practical implementation, a shorter circuit with a smaller number of inductances.

3 Rozrakhovuyemo postyni reworking of the filter.

, mH

, nF

4 For an ideal filter with an oscillator support of 1 ohm, an input support of 1 ohm,
the table of normalization coefficients of the Butterworth filter has been compiled. At the skin row, the tables of coefficients are symmetrical, increase to the middle, and then change.

5 In order to know the elements of the scheme, it is necessary to multiply the transformation by the coefficient of the table.

Expand the parameters of the Butterworth low-pass filter, for example PP = 0.15 kHz, =25 kHz, =30 dB,
=75 Ohm. Know
for three points.

29.3 HPF Butterworth.

Filters HPF - tse chotiripoles, for those in the range (
) is small, but in the range (
) is large, so that the filter is guilty of passing high-frequency jets into the navigation.

Shards of HPF are to blame for passing streams of high frequencies, then the path of the stream, which is going to be motivated, is to blame for the frequency deposited element, which is good for passing streams of high frequencies and badly streams of low frequencies. Such an element is a capacitor.

F
HF T-like

HPF P-like

Install the capacitor in sequence with the voltages, shards
the same frequency increase
changes, so high-frequency jets can easily pass at the input through the capacitor. Install the inductance coil in parallel with the voltage, shards
and with the increase in frequency increase
Therefore, streams of low frequencies flicker through the inductance and do not interfere with the input.

The Butterworth HPF analysis is similar to the Butterworth LPF analysis, carried out using the same formulas, only

.

Expand the high-pass filter of Butterworth HPF, as well as
Ohm
kHz,
db,
kHz. Know:
.

Lesson topic 30: Related and notch filters Butterworth.

Significant part of the theory of the development of digital BIX-filters (that is, filters from the non-stopped impulse response) is based on reasonable methods for the development of filters for an 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. Детальний аналіз переваг та недоліків способів апроксимації заданих характеристик, відповідних цим фільтрам, можна знайти в ряді робіт, присвячених методам розрахунку аналогових фільтрів, тому нижче будуть лише коротко перераховані основні властивості фільтрів кожного типу та наведені розрахункові співвідношення, необхідні для отримання коефіцієнтів аналогових фільтрів.

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 a function is approximated, as a rule, there is a square of the amplitude characteristic (bessel filter). 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 changes in amplitude characteristics on 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 lion's piv area 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 should be calculated from the mind of 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

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

Rice. 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.

In these articles, we will talk about the Butterworth filter, look at the orders of filters, decade and octave, and examine in detail the third-order low-pass filter with a rozrachunk and scheme.

Entry

In attachments, like vikoristovuyut filters for shaping the frequency spectrum of the signal, for example, in communication systems or control, the shape or the width of the recession is also called the “smooth transition”, for a simple filter of the first order, you can use a long time or you need a wide active filter, razroblenі z more nizh one "zamovlennyam". Qi types of filters sound like filters of a high order or n-th order.

Filter order

The foldability or the type of the filter is determined by the "order" of the filters and the number of reactive components, such as capacitors or coils of inductance of the first design. We also know that the rate of decline and, therefore, the width of the smog transition can be found according to the serial number of the filter and that for a simple filter of the first order, the standard rate of decline is 20 dB/decade or 6 dB/octave.

Then for the filter, which may have the n-th serial number, in the meantime, there will be a slowdown of 20n dB/decade or 6n dB/octave. In this manner:

• first order filter maximum rolloff 20 dB/decade (6 dB/octave)
• filter in a different order Sweep rate 40 dB/decade (12 dB/octave)
• fourth order filter May have a rolloff frequency of 80 dB/decade (24 dB/octave) too thin.

Filters of a high order, such as the third, fourth and fifth, are formed by a path of cascading single filters of the first and other order.

For example, two low-pass filters of a different order can be cascaded to remove a fourth-order low-pass filter, and so on. Irrespective of those that the order of the filter, which may be formed, not obmezheniya, zі zbіlshennyam order zbіlshuyusya yogo rozіrі varіst, and kozhu zhuєєєєєєєєєєє іtіknіstі.

Decadi and octavi

The rest of the comment about Decadesі Octaves. Beyond the frequency scale decade- This is a tenfold increase (multiplied by 10) or a tenfold change (divided by 10). For example, 2 to 20 Hz represent one decade, while 50 to 5000 Hz represent two decades (50 to 500 Hz, then 500 to 5000 Hz).

Octave- tse subwar (multiply by 2) or change vdvіchі (divided by 2) beyond the frequency scale. For example, 10 to 20 Hz represents one octave, and 2 to 16 Hz represents three octaves (2 to 4, 4 to 8 i, nareshti, 8 to 16 Hz), changing the frequency. In whatever mood, logarithmic the scales are widely varied in the frequency range for determining the value of the frequency when working with subsidiaries and filters, so it is important to understand them.

Oskіlki resistors, which determine the frequency, all are equal, as well as capacitors, which determine the frequency, or the cut frequency (C) for the first, second, third, or wind for the fourth-order filter, also due to buti equal and known, vikoristovuyuyuchi know the equalization:

As for the filters of the first and other order, the high-frequency filters of the third and fourth order are formed by simply mutually exchanging the position of the initial frequency of the components (resistors and capacitors) in an equivalent low-frequency filter. High-order filters can be designed by adding procedures, as we have previously worked out in the low-pass and high-pass filters. However, the high coefficient of the filter strength of the high order is fixed, oskіlki all components that determine the frequency, however.

Filter Approximations

We examined low-frequency and high-frequency filter circuits of the first order, their resulting frequency phase characteristics. The ideal filter has given us the specifics of maximum smog flow rate and flatness, minimum smog flow rate, as well as steeper smog flow rate, so that the smog drop (smuga transition) will increase, and it is obvious that a large number of merezhevy vіdgukіv will please you.

It is not surprising that in the linear design of analog filters there are a number of approximation functions, in which there is a mathematical approach for the best approximation of the transfer function, which is necessary for us to design filters.

Such constructions look like Eliptic, Butterworth, Chebishiv, Bessil, Cower and many others. 3 of the five "classic" functions of approximation of a linear analog filter only Butterworth filter and especially construction low-pass Butterworth filter will be considered here as a function, which is the most victorious.

Low pass Butterworth filter

Frequency response of the approximation function Butterworth filter Also often referred to as the “maximally flat” (no ripple) response, the smoky bandwidth is designed in such a way that the mother’s frequency response is as flat as possible, and mathematically it’s possible from 0 Hz (DC) to a frequency of -3 dB without ripple. Larger high frequencies beyond the boundaries of the cutoff point are reduced to zero in the smooth bands at a level of 20 dB/decade or 6 dB/octave. The reason that the “factor of quality”, “Q” is only 0.707.

However, one of the main shortcomings of the Butterworth filter is those that reach the width of the smog flow for a broad smog transition, if the filter is changed from a smug flow to a smug of fluff. Vin also has poor phase characteristics. The ideal frequency response is called the "smooth wall" filter, which is the standard Butterworth approximation for different orders of the filter lower.

To pay attention, the greater the order of the Butterworth filter, the greater the number of cascading slabs in the filter design and the closer the filter comes to the ideal result of the “single stone”.

However, in practice, the ideal frequency response of Butterworth is unattainable, the shards out there call out to the transcendental pulsation of the smog of transmission.

More specifically, as the Butterworth filter represents the "n-th" order, the frequency response is given as:

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

Even though A max is assigned at a frequency that is more common at the top of the line -3 dB (c), then ε will be more equal to 1, and ε 2 will also be more equal to 1. However, if you now want to calculate A max with the other value of the force behind the voltage, for example, 1 dB or 1.1220 (1 dB = 20 * logA max), then the same value of ε is found behind this formula:

Submitting data to equal, we will take:

frequency response filter can be calculated mathematically transfer function zі the standard of transmission of the voltage Function H (jω) and recorded at the sight:

Note: (jω) can also be written as (s) for recognition S-region. and the resulting transfer function for the low-pass filter of a different order is shown as:

Normalization of low-pass Butterworth filter polynomials

In order to help with the development of its low-frequency filters, Butterworth created standard tables of normalization of low-frequency polynomials of a different order with the adjusted value of the coefficient, which would indicate a frequency of 1 radian / s.

Rozrahunok that low-pass Butterworth filter circuit

Find the order of the active low-pass Butterworth filter, whose induction characteristics are: A max = 0.5 dB at the bandwidth frequency (ωp) of 200 radian/sec (31.8 Hz), and A min = -20 dB at the smog frequency of the grain (ωs ) 800 rad/sec. Also, we will develop the scheme of the Butterworth filter, so that we can confirm it.

First, the maximum strength of the smug transmission A max = 0.5 dB, as a more powerful strength 1,0593 , Remember that: 0.5 dB = 20 * log (A) at a frequency (ωp) of 200 rad / s, so the value of epsilon ε can be found from:

In a different way, the minimum strength of the zupinka smog A min = -20 dB, as a more powerful strength 10 (-20 dB = 20*log(A)) at the flute frequency (ωs) 800 rad/s or 127.3 Hz.

Substituting the value for the high frequency response of the Butterworth filters gives us the following:

So since n can be a whole number, then we will come to the largest values ​​of 2.42 will be n = 3 to that "Required filter of the third order", the one for the fold Butterworth filter third order, filter stage of another order is required cascading 3 step of the filter of the first order.

From the above table of normalizing low-frequency Butterworth polynomials, the third-order filter coefficient is given as (1 + s) (1 + s + s 2), and then gives us the power 3-A = 1 or A = 2. V A \u003d 1 + (Rf / R1), variable value for the resistor zvorotny zv'azku Rf i resistor R1 gives us the value of 1 kOhm and 1 kOhm, obviously, yak: (1 kOhm / 1 kOhm) + 1 = 2.

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

In this way, the frequency of the cut is set as 284 rad / s or 45.2 Hz (284 / 2π), and, knowing the formula 1 / RC, we can know the value of the resistors and capacitors for our third-order circuit.

Please note that the closest gain to 0.352 uF will be 0.36 uF or 360 nF.

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

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