Lesson Topic 28: Classification of Electrical Filters.

28.1 Definitions.

An electric frequency filter is called a four-port device, which passes currents of some frequencies well with low attenuation (3 dB attenuation), and currents of other frequencies poorly with a large attenuation (30 dB).

The range of frequencies in which the attenuation is small is called the bandwidth.

The range of frequencies in which the attenuation is large is called the stopband.

A transition strip is introduced between these strips.

The main characteristic of electrical filters is the frequency dependence of the operating attenuation.

This characteristic is called the attenuation frequency response.

- the cutoff frequency at which the operating attenuation is 3 dB.

- permissible attenuation, set by the mechanical parameters of the filter.

- admissible frequency corresponding to admissible attenuation.

PP - bandwidth - the frequency range in which
dB.

PZ - stop band - the frequency range in which the operating attenuation is greater than the allowable one.

28.2 Classification

1
By bandwidth location:

a) LPF - low-pass filter - passes the low frequencies and delays the high ones.

It is used in communication equipment (television receivers).

b
) HPF - high-pass filter - passes high frequencies and delays low ones.

v
) PF - band pass filters - pass only a certain frequency band.

G
) ZF - notch or blocking filters - do not pass only a certain frequency band, and the rest do.

2 By element base:

a) LC filters (passive)

b) RC filters (passive)

c) active ARC filters

d) special types of filters:

Piezoelectric

Magnetostrictive

3 By mathematical support:

a
) Butterworth filters. Operating attenuation characteristic
has a value of 0 at the frequency f = 0, and then increases monotonically. In the passband, it has a flat characteristic - this is an advantage, but in the stopband it is not steep - this is a disadvantage.

b) Chebyshev filters. To obtain a steeper characteristic, Chebyshev filters are used, but they have "waviness" in the passband, which is a disadvantage.

c) Zolotarev filters. Operating attenuation characteristic
in the passband it has waviness, and in the stopband it has a dip in the characteristics.

Lesson Topic 29: Butterworth Low and High Pass Filters.

29.1 Butterworth LPF.

Butterworth proposed the following attenuation formula:

, dB

where
- Butterworth function (normalized frequency)

n - filter order

For LPF
, where - any desired frequency

- cutoff frequency, which is

To realize this characteristic, L&C filters are used.

AND

inductance is placed in series with the load, since
and with growth increases
Therefore, currents of low frequencies will easily pass through the resistance of the inductor, while currents of high frequencies will be delayed and will not enter the load.

The capacitor is placed in parallel with the load, since
, therefore, the capacitor passes high-frequency currents well and low-frequency currents well. The high-frequency currents are closed through the capacitor, and the low-frequency currents will pass into the load.

The filter circuit consists of alternating L&C.

3rd order Butterworth LPF T-shaped

LPF Butterworth. 3rd order U-shaped.

Plan:

Introduction

• 1 Overview
  • 1.1 Normalized Butterworth polynomials
  • 1.2 Maximum smoothness
  • 1.3 Slope at high frequencies
• 2 Filter design
  • 2.1 Cauer topology
  • 2.2 Sullen-Kay topology
• 3 Comparison with other line filters
• 4 Example
Literature

Introduction

Butterworth filter- one of the types of electronic filters. Filters of this class differ from others in their design method. The Butterworth filter is designed so that its frequency response is as smooth as possible at the passband frequencies.

Such filters were first described by the British engineer Stephen Butterworth in the article "On the theory of filter amplifiers" (eng. On the Theory of Filter Amplifiers ), In the magazine Wireless Engineer in 1930.

1. Overview

The frequency response of the Butterworth filter is as smooth as possible at the passband frequencies and drops to almost zero at the frequencies of the suppression band. When displaying the frequency response of the Butterworth filter on a logarithmic frequency response, the amplitude decreases towards minus infinity at the frequencies of the stopband. In the case of a first-order filter, the frequency response decays at a rate of −6 decibels per octave (-20 decibels per decade) (in fact, all first-order filters, regardless of type, are identical and have the same frequency response). For a second order Butterworth filter, the frequency response attenuates by −12 dB per octave, for a third order filter, by −18 dB, and so on. The frequency response of the Butterworth filter is a monotonically decreasing function of frequency. The Butterworth filter is the only filter that preserves the frequency response for higher orders (with the exception of a steeper roll-off on the suppression band), while many other types of filters (Bessel filter, Chebyshev filter, elliptical filter) have different frequency response shapes at different orders.

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

Frequency response for low-pass Butterworth filters of order from 1 to 5. Characteristic slope - 20 n dB / decade, where n- filter order.

As with all filters, when considering the frequency characteristics, a low-pass filter is used, from which a high-pass filter can be easily obtained, and, by turning on several such filters in series, a band-pass filter or a notch filter.

The frequency response of the th order Butterworth filter can be obtained from the transfer function:

It is easy to see that for infinite values ​​the frequency response becomes a rectangular function, and frequencies below the cutoff frequency will be skipped with a gain, and frequencies above the cutoff frequency will be completely suppressed. For finite values, the slope of the characteristic will be flat.

With the help of a formal replacement, we represent the expression in the form:

The poles of the transfer function are located on a circle of radius equidistant from each other in the left half-plane. That is, the transfer function of the Butterworth filter can be determined only by determining the poles of its transfer function in the left half-plane of the s-plane. The th pole is determined from the following expression:

The transfer function can be written as:

Similar reasoning is applicable to digital Butterworth filters, with the only difference that the ratios are not written for s-plane, and for z-planes.

The denominator of this transfer function is called the Butterworth polynomial.

1.1. Normalized Butterworth polynomials

Butterworth polynomials can be written in complex form, as shown above, but they are usually written as ratios with real coefficients (complex conjugate pairs are combined using multiplication). Cutoff frequency polynomials are normalized:. The normalized Butterworth polynomials thus have the following canonical form:

, - even, - odd

Below are the coefficients of the Butterworth polynomials for the first eight orders:

Polynomial coefficients 1 2 3 4 5 6 7 8

1.2. Maximum smoothness

Having accepted and, the derivative of the amplitude characteristic with respect to frequency will look like this:

It decreases monotonically for all, since the gain is always positive. Thus, the frequency response of the Butterworth filter has no ripple. When expanding the amplitude characteristic in a series, we get:

In other words, all derivatives of the amplitude-frequency characteristic with respect to frequency up to 2 n-th are equal to zero, which implies "maximum smoothness".

1.3. Slope at high frequencies

Having accepted, we find the slope of the frequency response logarithm at high frequencies:

In decibels, the high frequency asymptote has a slope of −20 n dB / decade.

2. Filter design

There are a number of different filter topologies that implement analog linear filters. These schemes differ only in the values ​​of the elements, the structure remains unchanged.

2.1. Cauer topology

Cauer topology uses passive elements (capacitance and inductance). A Butteworth filter with a given transfer function can be constructed in a Cauer type 1 form. The kth filter element is given by:

; k is odd; k is even

2.2. Sullen-Kay topology

The Sullen-Kay topology uses active elements (operational amplifiers and capacitors) in addition to passive ones. Each stage of the Sullen-Kay circuit is a part of the filter, mathematically described by a pair of complex conjugate poles. The entire filter is obtained by connecting all stages in series. If a real pole comes across, it must be implemented separately, usually in the form of an RC-chain, and included in the overall circuit.

The transfer function of each stage in the Sallen-Kay circuit is:

The denominator needs to be one of the factors of the Butterworth polynomial. Having accepted, we get:

The last relation gives two unknowns that can be chosen arbitrarily.

3. Comparison with other line filters

The figure below shows the frequency response of the Butterworth filter in comparison with other popular line filters of the same (fifth) order:

The figure shows that the Butterworth filter rolloff is the slowest of the four, but it also has the smoothest frequency response at the passband frequencies.

4. Example

An analog low-pass Butterworth filter (Cauer topology) with a cutoff frequency with the following element ratings: farad, ohm, and henry.

Logarithmic graph of the density of the transfer function H (s) on the plane of the complex argument for a third-order Butterworth filter with a cutoff frequency. Three poles lie on a circle of unit radius in the left half-plane.

Consider a third-order analog low-pass Butterworth filter with farad, ohm, and henry. Designating the impedance of the containers C how 1 / Cs and impedance of inductors L how Ls, where is a complex variable, and using the equations for calculating electrical circuits, we obtain the following transfer function for such a filter:

The frequency response is given by the equation:

and the phase-frequency characteristic is given by the equation:

Group delay is defined as the minus derivative of the phase with respect to the circular frequency and is a measure of the distortion of the signal with respect to phase at various frequencies. The logarithmic frequency response of such a filter has no ripple either in the passband or in the suppression band.

The plot of the modulus of the transfer function on the complex plane clearly indicates three poles in the left half-plane. The transfer function is completely determined by the arrangement of these poles on the unit circle symmetrically about the real axis.

Replacing each inductance with a capacitor, and the capacitances with inductors, we get a high-frequency Butterworth filter.

And the group delay of a third-order Butterworth filter with a cutoff frequency

Literature

• V.A. Lucas Automatic control theory. - M .: Nedra, 1990.
• B.Kh. Krivitsky Handbook on the theoretical foundations of radio electronics. - M .: Energy, 1977.
• Miroslav D. Lutovac Filter Design for Signal Processing using MATLAB © and Mathematica ©. - New Jersey, USA .: Prentice Hall, 2001 .-- ISBN 0-201-36130-2
• Richard W. Daniels Approximation Methods for Electronic Filter Design. - New York: McGraw-Hill, 1974 .-- ISBN 0-07-015308-6
• Steven W. Smith The Scientist and Engineer's Guide to Digital Signal Processing. - Second Edition. - San-Diego: California Technical Publishing, 1999. - ISBN 0-9660176-4-1
• Britton C. Rorabaugh Approximation Methods for Electronic Filter Design. - New York: McGraw-Hill, 1999 .-- ISBN 0-07-054004-7
• B. Widrow, S.D. Stearns Adaptive Signal Processing. - Paramus, NJ: Prentice-Hall, 1985 .-- ISBN 0-13-004029-0
• S. Haykin Adaptive Filter Theory. - 4rd Edition. - Paramus, NJ: Prentice-Hall, 2001 .-- ISBN 0-13-090126-1
• Michael L. Honig, David G. Messerschmitt Adaptive Filters - Structures, Algorithms, and Applications. - Hingham, MA: Kluwer Academic Publishers, 1984 .-- ISBN 0-89838-163-0
• J.D. Markel, A.H. Gray, Jr. Linear Prediction of Speech. - New York: Springer-Verlag, 1982 .-- ISBN 0-387-07563-1
• L.R. Rabiner, R.W. Schafer Digital Processing of Speech Signals. - Paramus, NJ: Prentice-Hall, 1978 .-- ISBN 0-13-213603-1
• Richard J. Higgins Digital Signal Processing in VLSI. - Paramus, NJ: Prentice-Hall, 1990 .-- ISBN 0-13-212887-X
• A. V. Oppenheim, R. W. Schafer Digital Signal Processing. - Paramus, NJ: Prentice-Hall, 1975. - ISBN 0-13-214635-5
• L. R. Rabiner, B. Gold Theory and Application of Digital Signal Processing. - Paramus, NJ: Prentice-Hall, 1986 .-- ISBN 0-13-914101-4
• John G. Proakis, Dimitris G. Manolakis Introduction to Digital Signal Processing. - Paramus, NJ: Prentice-Hall, 1988 .-- ISBN 0-02-396815-X

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> LPF1)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> HPF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> PF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> RF)

4th order Butterworth filter

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> LPF1)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> HPF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> PF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> RF)

3rd order Chebyshev filter

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> LPF1)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> HPF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> PF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> RF)

4-order Chebyshev filter

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> LPF1)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> HPF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> PF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> RF)

3rd order Bessel filter

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> LPF1)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> HPF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> PF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> RF)

Bessel filter of the 4th order

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> LPF1)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> HPF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> PF)

CONVERSION OF FREQUENCY PROPERTIES OF ZF (LPF -> RF)

Analyze the effect of errors in setting the coefficients of the digital low-pass filter on the frequency response (changing one of the coefficients b j). Describe the nature of the frequency response. Make a conclusion about the influence of changing one of the coefficients on the filter's behavior.

The analysis of the influence of errors in setting the coefficients of the digital low-pass filter on the frequency response will be carried out using the example of a Bessel filter of the 4th order.

Let us choose the value of the deviation of the coefficients ε equal to –1.5% so that the maximum deviation of the frequency response is about 10%.

The frequency response of the "ideal" filter and filters with modified coefficients by the value ε is shown in the figure:

AND

The figure shows that the greatest influence on the frequency response is exerted by the change in the coefficients b 1 and b 2, (their value exceeds the value of other coefficients). Using a negative value of ε, we note that positive coefficients decrease the amplitude in the lower part of the spectrum, while negative ones increase it. With a positive value of ε, everything happens the other way around.

Quantize the coefficients of the digital filter for such a number of binary digits so that the maximum deviation of the frequency response from the original is about 10 - 20%. Sketch the frequency response and describe the nature of its change.

By changing the number of digits of the fractional part of the coefficients b j Note that the maximum deviation of the frequency response from the initial one, not exceeding 20%, is obtained at n≥3.

Frequency response type at different n shown in the figures:

n = 3, maximum deviation of frequency response = 19.7%

n = 4, maximum deviation of frequency response = 13.2%

n = 5, maximum frequency response deviation = 5.8%

n = 6, maximum deviation of frequency response = 1.7%

Thus, it can be noted that an increase in the bit depth when quantizing the filter coefficients leads to the fact that the frequency response of the filter tends more and more to the original one. However, it should be noted that this complicates the physical realizability of the filter.

Quantization for different n can be traced in the figure:

Institute of Non-Ferrous Metals and Gold of Siberian Federal University

Department of Automation of Production Processes

Filter types  LPF Butterworth  LPF Chebyshev I type  Minimum filter order  LPF with MOS 

 LPF on INUN  Biquad LPF  Configuring 2nd order filters  Odd-order low-pass filter 

 LPF Chebyshev II type  Elliptical LPF  Elliptical LPF on INUN  Elliptical low-pass filter on 3 capacitors  Biquad Elliptical LPF  Setting the Chebyshev LPF II type and elliptical 

 Configuring 2nd order filters  All pass filters  LPF Modeling  Creating diagrams 

 Calculation of transitional x-k  Calculation of frequency x-k  Completing of the work  Control questions 

Laboratory work No. 1

"Exploring signal filtering in a Micro-Cap 6/7 environment"

purpose of work

1. Examine the main types and characteristics of filters

2. Explore the simulation of filters in the Micro-Cap 6 environment.

3. Investigate the characteristics of active filters in Micro-Cap 6 environment

Theoretical information

1. Types and characteristics of filters

Signal filtering plays an important role in digital control systems. In them, filters are used to eliminate random measurement errors (superposition of interference signals, noise) (Fig. 1.1). Distinguish between hardware (circuit) and digital (software) filtering. In the first case, electronic filters from passive and active elements are used, in the second case, various software methods for isolating and eliminating interference are used. Hardware filtering is used in modules of USO (communication devices with an object) of controllers and distributed data acquisition and control systems.

Digital filtering is used in the upper-level UHM of the APCS. This paper discusses in detail the issues of hardware filtering.

There are the following types of filters:

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

high-pass filters (pass high frequencies and delay low frequencies);

bandpass filters (pass a band of frequencies and delay frequencies above and below this band);

bandstop filters (which delay a bandwidth and allow frequencies above and below that bandwidth to pass).

The transfer function (TF) of the filter has the form:

where ½ N(j w) ½- module PF or frequency response; j (w) - phase-frequency characteristic; w - angular frequency (rad / s) associated with the frequency f (Hz) by the relation w = 2p f.

П Ф of the implemented filter has the form

where a and b - constants, and T , n = 1, 2, 3 ... (m £ n).

Denominator polynomial degree n determines the order of the filter. The higher it is, the better the frequency response, but the circuit is more complex, and the cost is higher.

The frequency ranges or bands in which the signals pass are the passbands and in them the frequency response value is ½ N(j w) ½ is large, and ideally constant. The frequency ranges in which the signals are suppressed are stopbands and in them the frequency response is small, and in the ideal case is equal to zero.

The frequency response of real filters differs from the theoretical frequency response. For a low-pass filter, the ideal and real frequency response are shown in Fig. 1.6.

In real filters, the passband is the frequency range (0 -  s), where the frequency response is greater than the specified value A 1 . Retention lane - this is the frequency range ( 1 -∞), in which the frequency response is less than the value - A 2 . The frequency interval of the transition from the passband to the stopband, ( c - 1) is called the transition region.

Often, attenuation is used instead of amplitude to characterize filters. Attenuation in decibels (dB) is determined by the formula

Amplitude value А = 1 corresponds to attenuation a= 0. If A 1 = A /
= 1/= 0.707, then the attenuation at the frequency w c:

The ideal and real characteristics of a low-pass filter using attenuation are shown in Fig. 1.7.

Rice. 1.8. LPF ( a) and its 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 inductors are unsatisfactory: large dimensions and deviations from ideal characteristics. Inductors are poorly suited for integral performance. The simplest low-pass filter (LPF) and its frequency response are shown in Fig. 1.8.

Active filters are created based on R, C elements and active elements - operational amplifiers (OA). The op-amp must have: high gain (50 times more than that of the filter); high slew rate of the output voltage (up to 100-1000 V / μs).

Rice. 1.9. T- and U-shaped LPF

Active low-pass filters of the first and second orders are shown in Fig. 1.10 - 1.11. Building filters n-th order is carried out by cascading links N 1 , N 2 , ... , N m with PF N 1 (s), H 2 (s), ..., H m ( s).

Even order filter with NS > 2 contains n/ 2 links of the second order, connected in cascade. Odd order filter with NS > 2 contains ( NS - 1) / 2 links of the second order and one link of the first order.

For first-order filters PF

where V and WITH - constant numbers; P(s) is a polynomial of degree two or less.

LPF has maximum attenuation in the passband a 1 does not exceed 3 dB, and the attenuation in the stopband a 2 is in the range from 20 to 100 dB. LPF gain is the value of its transfer function at s = 0 or the value of its frequency response at w = 0 , i.e. . is equal to A.

There are the following types of LPF:

Butterworth- have a monotonic frequency response (Fig. 1.12);

Chebyshev (type I) - frequency response contains ripple in the passband and is monotonic in the stopband (Fig. 1.13);

inverse Chebyshev(like II) - the frequency response is monotonic in the passband and has ripple in the stopband (Fig. 1.14);

elliptical - The frequency response has ripples both in the passband and in the stopband (Fig. 1.15).

Low Pass Butterworth Filter n-th order has a frequency response of the following form

The PF of the Butterworth filter as a polynomial filter is

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

where the parameters e and TO - constant numbers, and WITH NS- Chebyshev polynomial of the first kind of degree NS equal to

Swing R p can be reduced by choosing the value of the parameter e small enough.

The minimum acceptable passband attenuation - constant peak-to-peak ripple - is expressed in decibels as

.

The FS of the Chebyshev and Butterworth low-pass filters are identical in form and are described by expressions (1.15) - (1.16). The frequency response of the Chebyshev filter is better than the frequency response of the Butterworth filter of the same order, since the first has the width of the transition region. However, the PFC of the Chebyshev filter is worse (more nonlinear) than the PFC of the Butterworth filter.

The frequency response of the Chebyshev filter of this order is better than the frequency response of Butterworth, since the Chebyshev filter has a narrower transition region. However, the phase response of the Chebyshev filter is worse (more nonlinear) than the phase response of the Butterworth filter.

Phase characteristics of the Chebyshev filter for the 2-7th orders are shown in Fig. 1.18. For comparison, Fig. 1.18, the dashed line shows the phase response of the sixth-order Butterworth filter. It can also be noted that the phase response of high-order Chebyshev filters is worse than the phase response of lower-order filters. This is consistent with the fact that the frequency response of the high order Chebyshev filter is better than the frequency response of the lower order filter.

1.1. SELECTING THE MINIMUM FILTER ORDER

Based on Fig. 1.8 and 1.9, we can conclude that the higher the order of the Butterworth and Chebyshev filters, the better their frequency response. However, the higher order complicates the circuit implementation and therefore increases the cost. Thus, it is important to select the minimum required filter order that satisfies the given requirements.

Let in the one shown in Fig. 1.2 the general characteristic specifies the maximum allowable attenuation in the passband a 1 (dB), the minimum allowable attenuation in the stopband a 2 (dB), cutoff frequency w s (rad / s) or f c (Hz) and the maximum permissible width of the transition region T W, which is defined as follows:

where logarithms can be either natural or decimal.

Equation (1.24) can be written as

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

and substitute the obtained relation into (1.25) to find the dependence of the order NS on the width of the transition region, and not on the frequency w 1. Parameter T W / w s called normalized the width of the transition region and is a dimensionless quantity. Hence, T W and w c can be specified in both radians per second and hertz.

In a similar way, based on (1.18) for K = 1 find the minimum order of the Chebyshev filter

and it follows from (1.25) that a Butterworth filter satisfying these requirements must have the following minimum order:

Finding the next largest integer again, we get NS= 4.

This example clearly illustrates the advantage of the Chebyshev filter over the Butterworth filter, if the main parameter is the frequency response. In the case under consideration, the Chebyshev filter provides the same transfer function slope as the double complexity Butterworth filter.

1.2. LPF WITH MULTI-LOOP FEEDBACK

AND INFINITE GAIN COEFFICIENT

Rice. 1.11. LPF with MOS second order

There are many ways to build active Butterworth and Chebyshev LPFs. Below we will consider some of the most commonly used general circuits, starting with the simple ones (in terms of the number of circuit elements required) and moving on to the most complex ones.

For higher-order filters, equation (1.29) describes the TF of a typical second-order link, where TO - its gain; V and WITH - link coefficients given in the reference literature. One of the simplest schemes of active filters that implement the low-pass PF according to (1.29) is shown in Fig. 1.11.

This scheme realizes equation (1.29) with inverting gain - TO(TO> 0) and

Resistances satisfying equation (1.30) are

A sensible approach is to set the capacitance to a nominal value C 2, close to 10 / f c μF and select the largest available capacitance rating C 1 satisfying equation (1.31). Resistances should be close to the values ​​calculated by (1.31). The higher the filter order, the more critical these requirements are. If calculated resistance ratings are not available, it should be noted that all resistance values ​​can be multiplied by a common factor, provided that the capacitance values ​​are divided by the same factor.

As an example, suppose you want to design a second-order MOC Chebyshev filter with 0.5 dB flatness, 1000 Hz bandwidth, and a gain of 2. In this case TO= 2, w c = 2π (1000), and from Appendix A we find that B = 1.425625 and C = 1.516203. Choosing the nominal value C 2 = 10/f c= 10/1000 = 0.01 μF = 10 -8 F, from (1.32) we obtain

Now suppose you want to design a sixth order Butterworth filter with MOC, cutoff frequency f c= 1000 Hz and gain K = 8. It will consist of three second-order links, each with a TF determined by equation (2.1). Let's choose the gain of each link K= 2, which provides the required gain of the filter itself 2 ∙ 2 ∙ 2 = 8. From Appendix A for the first link we find V= 0.517638 and C = 1. Select the nominal capacitance value again WITH 2 = 0.01 μF and in this case from (2.21) we find WITH 1 = 0.00022 μF. Let's set the nominal value of the capacitance WITH 1 = 200 pF and from (2.20) we find the resistance values R 2 = 139.4 kΩ; R 1 = 69.7 kΩ; R 3 = 90.9 kΩ. The other two links are calculated in a similar way, and then the links are cascaded to create a sixth order Butterworth filter.

Because of its relative simplicity, the MOC filter is one of the most popular types of inverting gain filters. It also has certain advantages, namely good stability of characteristics and low output impedance; thus, it can be immediately cascaded with other links to implement a higher order filter. The disadvantage of the circuit is that it is impossible to achieve a high value of the quality factor Q without a significant scatter in the values ​​of the elements and high sensitivity to their change. For good results, the gain is TO

Adjusted LPF-filter. ... Moe-structure, is the ability to adjust the gain and band filter when changing denominations minimum ... filter on microcircuits type... has the same order quantities as ... classical filtersChebyshev and Butterworth, ...

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

where
- Butterworth function corresponding to the permissible frequency .

- permissible attenuation.

2 We draw the filter circuit of the obtained order. For practical implementation, circuits with fewer inductors are preferred.

3 We calculate the constant transformations of the filter.

, mH

, nF

4 For an ideal filter with a generator impedance of 1 ohm, a load impedance of 1 ohm,
compiled a table of normalized Butterworth filter coefficients. In each row of the table, the coefficients are symmetric, increase towards the middle and then decrease.

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 HFH.

HPF filters are four-port networks, in which in the range (
), the attenuation is small, and in the range (
) - large, that is, the filter must pass high-frequency currents into the load.

Since the HPF must pass high-frequency currents, there must be a frequency-dependent element in the path of the current going to the load, which passes high-frequency currents well and low-frequency currents poorly. This element is a capacitor.

F
HF T-shaped

HPF U-shaped

The capacitor is placed in series with the load, since
and with increasing frequency
decreases, therefore, high-frequency currents easily pass into the load through the capacitor. The inductor is placed in parallel with the load, since
and as the frequency increases,
, therefore, currents of low frequencies are closed through inductors and will not enter the load.

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

.

Calculate the Butterworth HPF high-pass filter if
Ohm,
kHz,
dB,
kHz. Find:
.

Lesson 30: Butterworth Bandpass and Notch Filters.