MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE

Kharkov National University of Radio Electronics

Department of REU

COURSE WORK

CALCULATION AND EXPLANATORY NOTE

BUTTERWORTH HIGH PASS FILTER

Kharkov 2008

Technical task

Design a high-pass filter (HPF) with approximation of the amplitude-frequency response (AFC) by a Butterworth polynomial, determine the required filter order if the AFC parameters are specified (Fig. 1): K 0 = 26 dB

U m In =250mV

where is the maximum transmission coefficient of the filter;

Minimum transmission coefficient in the passband;

Maximum filter gain in the delay band;

Cutoff frequency;

The frequency from which the filter gain is less.

Figure 1 – Butterworth high-pass filter pattern.

Provide slight sensitivity to deviations in element values.

ABSTRACT

Settlement and explanatory note: 26 pp., 11 figures, 6 tables.

Purpose of the work: synthesis of an active RC high-pass filter circuit and calculation of its components.

Research method: approximation of the frequency response of the filter by the Butterworth polynomial.

The approximated transfer function is implemented using an active filter. The filter is built by a cascade connection of independent links. Active filters use non-inverting finite gain amplifiers, which are implemented using operational amplifiers.

The results of the work can be used to synthesize filters for radio engineering and household equipment.

Introduction

1. Review of similar schemes

3.1 Implementation of high-pass filter normalization

3.2 Determining the required filter order

3.3 Definition of Butterworth polynomial

3.4 Reverse transition from normalized to designed high-pass filter

3.5Transition from the transfer function to the circuit

3.6 Transition from the transfer function to the circuit

4. Calculation of circuit elements

5. Methodology for adjusting the developed filter

Introduction

Until recently, the results of comparing digital and analog devices in radio equipment and technical means of telecommunication could not but cause a feeling of dissatisfaction. Digital components, implemented with the widespread use of integrated circuits (ICs), were distinguished by their design and technological completeness. The situation was different with analog signal processing units, which, for example, in telecommunications accounted for 40 to 60% of the volume and weight of communication equipment. Bulky, containing a large number of unreliable and labor-intensive winding elements, they looked so depressing against the backdrop of large integrated circuits that they gave rise to the opinion of a number of experts about the need for “total digitalization” of electronic equipment.

The latter, however, like any other extreme, did not lead (and could not lead) to results adequate to those expected. The truth, as in all other cases, turned out to be somewhere in the middle. In some cases, equipment built on functional analog units, the elemental basis of which is adequate to the capabilities and limitations of microelectronics, turns out to be more effective.

Adequacy in this case can be ensured by the transition to active RC circuits, the elemental basis of which does not include inductors and transformers, which are fundamentally not implemented by microelectronics.

The validity of such a transition is currently determined, on the one hand, by the achievements of the theory of active RC circuits, and on the other, by the successes of microelectronics, which have provided developers with high-quality linear integrated circuits, including integrated operational amplifiers (OP-amps). These op-amps, having great functionality, have significantly enriched analog circuitry. This was especially evident in the circuitry of active filters.

Until the 60s, mainly passive elements were used to implement filters, i.e. inductors, capacitors and resistors. The main problem in implementing such filters is the size of the inductors (at low frequencies they become too bulky). With the development of integrated operational amplifiers in the 60s, a new direction in the design of active filters based on op-amps appeared. Active filters use resistors, capacitors and op-amps (active components), but do not have inductors. Subsequently, active filters almost completely replaced passive ones. Currently, passive filters are used only at high frequencies (above 1 MHz), outside the frequency range of most widely used op amps. But even in many high-frequency devices, such as radio transmitters and receivers, traditional RLC filters are being replaced by quartz and surface acoustic wave filters.

Nowadays, in many cases, analog filters are being replaced by digital ones. The operation of digital filters is ensured mainly by software, so they are much more flexible in use compared to analog ones. Using digital filters, it is possible to implement transfer functions that are very difficult to obtain using conventional methods. However, digital filters cannot yet replace analog filters in all situations, so the need for the most popular analog filters, active RC filters, remains.

1. Review of similar schemes

Filters are frequency-selective devices that pass or reject signals lying in certain frequency bands.

Filters can be classified according to their frequency characteristics:

1. Low-pass filters (LPF) - pass all oscillations with frequencies not higher than a certain cutoff frequency and a constant component.

2. High-pass filters (LPF) - pass all vibrations not lower than a certain cutoff frequency.

3. Bandpass filters (BPFs) – pass oscillations in a certain frequency band, which is determined by a certain level of frequency response.

4. Band-suppression filters (BPFs) - delay oscillations in a certain frequency band, which is determined by a certain level of frequency response.

5. Notch filters (RF) - a type of BPF that has a narrow delay band and is also called a plug filter.

6. Phase filters (PF) - ideally have a constant transmission coefficient at all frequencies and are designed to change the phase of input signals (in particular, for the time delay of signals).

Figure 1.1 – Main types of filters

Using active RC filters, it is impossible to obtain ideal shapes of frequency characteristics in the form of rectangles shown in Fig. 1.1 with a strictly constant gain in the passband, infinite attenuation in the suppression band and an infinite slope of the roll-off when moving from passband to suppression band. Designing an active filter is always a search for a compromise between the ideal form of the characteristic and the complexity of its implementation. This is called the “approximation problem.” In many cases, the requirements for filtration quality make it possible to get by with the simplest first- and second-order filters. Some circuits of such filters are presented below. Designing a filter in this case comes down to choosing a circuit with the most suitable configuration and subsequent calculation of the values ​​of element ratings for specific frequencies.

However, there are situations where the filtering requirements may be much more stringent, and higher order circuits than the first and second ones may be required. Designing high-order filters is a more complex task, which is the subject of this course work.

Below are some basic first-second order schemes with the advantages and disadvantages of each.

1. Low-pass filter-I and low-pass filter-I based on a non-inverting amplifier.

Figure 1.2 – Filters based on a non-inverting amplifier:

a) LPF-I, b) HPF-I.

The advantages of filter circuits include mainly ease of implementation and configuration, the disadvantages are low frequency response slope and low resistance to self-excitation.

2. Low-pass filter-II and low-pass filter-II with multi-loop feedback.

Figure 1.3 – Filters with multi-loop feedback:

a) LPF-II, b) HPF-II.

Table 2.1 – Advantages and disadvantages of low-pass filter-II with multi-loop feedback

Table 2.2 – Advantages and disadvantages of HPF-II with multi-loop feedback

2. LPF-II and HPF-IISallen-Kay.

Figure 1.4 – Sallen-Kay filters:

a) LPF-II, b) HPF-II

Table 2.3 – Advantages and disadvantages of Sallen-Kay low-pass filter-II.

Table 2.4 – Advantages and disadvantages of HPF-II Sallen-Kay.

3. LPF-II and HPF-II based on impedance converters.

Figure 1.5 – Low-pass filter II circuit based on impedance converters:

a) LPF-II, b) HPF-II.

Table 2.3 – Advantages and disadvantages of LPF-II and HPF-II based on impedance converters.

2. Selection and justification of the filter circuit

Filter design methods differ in design features. The design of passive RC filters is largely determined by the block diagram

Active AF filters are mathematically described by a transfer function. Frequency response types are given the names of transfer function polynomials. Each type of frequency response is implemented by a certain number of poles (RC circuits) in accordance with a given slope of the frequency response. The most famous are the approximations of Butterworth, Bessel, and Chebyshev.

The Butterworth filter has the most flat frequency response; in the suppression band, the slope of the transition section is 6 dB/oct per pole, but it has a nonlinear phase response; the input pulse voltage causes oscillation at the output, so the filter is used for continuous signals.

The Bessel filter has a linear phase response and a small steepness of the transition section of the frequency response. Signals of all frequencies in the passband have the same time delays, so it is suitable for filtering square wave pulses that need to be sent without distortion.

The Chebyshev filter is a filter of equal waves in the SP, a mass-flat shape outside it, suitable for continuous signals in cases where it is necessary to have a steep slope of the frequency response behind the cutoff frequency.

Simple first- and second-order filter circuits are used only when there are no strict requirements for filtration quality.

A cascade connection of filter sections is carried out if a filter order higher than the second is needed, that is, when it is necessary to form a transfer characteristic with a very large attenuation of signals in the suppressed band and a large attenuation slope of the frequency response. The resulting transfer function is obtained by multiplying the partial transfer coefficients

The circuits are built according to the same scheme, but the values ​​of the elements

R, C are different, and depend on the cutoff frequencies of the filter and its slats: f zr.f / f zr.l

However, it should be remembered that a cascade connection of, for example, two second-order Butterworth filters does not produce a fourth-order Butterworth filter, since the resulting filter will have a different cutoff frequency and a different frequency response. Therefore, it is necessary to select the coefficients of single links in such a way that the next product of transfer functions corresponds to the selected type of approximation. Therefore, designing an AF will cause difficulties in obtaining an ideal characteristic and the complexity of its implementation.

Thanks to the very large input and small output resistances of each link, the absence of distortion of the specified transfer function and the possibility of independent regulation of each link are ensured. The independence of the links makes it possible to widely regulate the properties of each link by changing its parameters.

In principle, it does not matter in which order the partial filters are placed, since the resulting transfer function will always be the same. However, there are various practical guidelines regarding the order in which partial filters must be connected. For example, to protect against self-excitation, a sequence of links should be organized in order of increasing partial limiting frequency. A different order can lead to self-excitation of the second link in the region of its frequency response surge, since filters with higher cutoff frequencies usually have a higher quality factor in the cutoff frequency region.

Another criterion is related to the requirements for minimizing the noise level at the input. In this case, the sequence of links is reversed, since the filter with the minimum limiting frequency attenuates the noise level that arises from the previous links of the cascade.

3. Topological model of the filter and voltage transfer function

3.1 In this paragraph, the order of the Butterworth high-pass filter will be selected and the type of its transfer function will be determined according to the parameters specified in the technical specifications:

Figure 2.1 – High-pass filter template according to the technical specifications.

Topological model of the filter.

3.2 Implementation of high-pass filter normalization

Based on the specification conditions, we find the boundary conditions of the filter frequency we need. And we normalize it by the transmission coefficient and by the frequency.

Behind the gear ratio:

K max =K 0 -K p =26-23=3dB

K min =K 0 -K z =26-(-5)=31dB

By frequency:

3.3 Determining the required filter order

Round n to the nearest integer value: n = 3.

Thus, to satisfy the requirements specified by the pattern, a third-order filter is needed.

3.4 Definition of Butterworth polynomial

According to the table of normalized transfer functions of Butterworth filters, we find the third-order Butterworth polynomial:

3.5 Reverse transition from normalized to designed high-pass filter

Let us carry out the reverse transition from the normalized high-pass filter to the designed high-pass filter.

· scaling by transmission coefficient:

Frequency scaling:

We make a replacement

As a result of scaling, we obtain the transfer function W(p) in the form:

Figure 2.2 – Frequency response of the designed Butterworth high-pass filter.

3.6 Transition from transfer function to circuit

Let us imagine the transfer function of the designed third-order high-pass filter as a product of the transfer functions of two active first- and second-order high-pass filters, i.e. as

And ,

where is the transmission coefficient at an infinitely high frequency;

– pole frequency;

– filter quality factor (the ratio of the gain at frequency to the gain in the passband).

This transition is fair, since the total order of active filters connected in series will be equal to the sum of the orders of individual filters (1 + 2 = 3).

The overall transmission coefficient of the filter (K0 = 19.952) will be determined by the product of the transmission coefficients of the individual filters (K1, K2).

Expanding the transfer function into quadratic factors, we obtain:

In this expression

. (2.5.1)

It is easy to notice that the pole frequencies and quality factors of the transfer functions are different.

For the first transfer function:

pole frequency;

The quality factor of the HPF-I is constant and equal to .

For the second transfer function:

pole frequency;

quality factor

In order for the operational amplifiers in each stage to be subject to approximately equal requirements for frequency properties, it is advisable to distribute the total transmission coefficient of the entire filter between each of the stages in inverse proportion to the quality factor of the corresponding stages, and select the maximum characteristic frequency (unity gain frequency of the op-amp) among all stages.

Since in this case the high-pass filter consists of two cascades, the above condition can be written as:

. (2.5.2)

Substituting expression (2.5.2) into (2.5.1), we obtain:

;

Let's check the correctness of calculation of transmission coefficients. The overall transmission coefficient of the filter in times will be determined by the product of the coefficients of the individual filters. Let's convert the IdB coefficient into several times:

Those. the calculations are correct.

Let's write down the transfer characteristic taking into account the values ​​calculated above ():

.

3.7 Selecting a third-order active high-pass filter circuit

Since, according to the task, it is necessary to ensure a slight sensitivity to deviations of the elements, we will choose as the first stage HPF-I based on a non-inverting amplifier (Fig. 1.2, b), and the second – HPF-II based on impedance converters (ICC), the diagram of which is shown in Fig. 1.5, b.

For HPF-I based on a non-inverting amplifier, the dependence of the filter parameters on the values ​​of the circuit elements is as follows:

For HPF-II based on KPS, the filter parameters depend on the nominal values ​​of the elements as follows:

; (3.4)

;

4. Calculation of circuit elements

· Calculation of the first stage (HPF I) with parameters

Let's choose R1 based on the requirements for the value of the input resistance (): R1 = 200 kOhm. Then from (3.2) it follows that

.

Let us choose R2 = 10 kOhm, then from (3.1) it follows that

· Calculation of the second stage (HPF II) with parameters

. .

Then (the coefficient in the numerator is selected so as to obtain the capacity rating from the standard E24 series). So C2 = 4.3 nF.

From (3.3) it follows that

From (3.1) it follows that

Let . So C1 = 36 nF.

Table 4.1 – Filter element ratings

From the data in Table 4.1 we can begin to model the filter circuit.

We do this using a special program Workbench5.0.

The simulation diagram and results are shown in Fig. 4.1. and Fig. 4.2, a-b.

Figure 4.1 – Third-order Butterworth high-pass filter circuit.

Figure 4.2 – Resulting frequency response (a) and phase response (b) of the filter.

5. Methodology for setting and regulating the developed filter

In order for a real filter to provide the desired frequency response, resistances and capacitances must be selected with great accuracy.

This is very easy to do for resistors, if they are taken with a tolerance of no more than 1%, and more difficult for capacitors, because their tolerances are in the region of 5-20%. Because of this, the capacitance is calculated first, and then the resistance of the resistors is calculated.

5.1 Selecting the type of capacitors

· We will choose a low-frequency type of capacitors due to their lower cost.

Small dimensions and weight of capacitors are required

· You need to choose capacitors with as little loss as possible (with a small dielectric loss tangent).

Some parameters of group K10-17 (taken from):

Dimensions, mm.

Weight, g0.5…2

Permissible deviation of capacity,%

Loss tangent0.0015

Insulation resistance, MOhm1000

Operating temperature range, – 60…+125

5.2 Selecting resistor type

· For the designed filter circuit, in order to ensure low temperature dependence, it is necessary to select resistors with a minimum TCR.

· The selected resistors must have a minimum intrinsic capacitance and inductance, so we will choose a non-wire type of resistors.

· However, non-wire resistors have a higher level of current noise, so it is also necessary to take into account the parameter of the self-noise level of the resistors.

Precision resistors type C2-29V meet the specified requirements (parameters taken from):

Rated power, W 0.125;

Range of nominal resistances, Ohm;

TKS (in the temperature range),

TKS (in the temperature range ),

Intrinsic noise level, µV/V1…5

Maximum operating voltage DC

and AC, V200

5.3 Selecting the type of operational amplifiers

· The main criterion when choosing an op-amp is its frequency properties, since real op-amps have a finite bandwidth. In order for the frequency properties of the op-amp not to affect the characteristics of the designed filter, it is necessary that for the unity gain frequency of the op-amp in the i-th stage the following relation is satisfied:

For the first cascade: .

For the second cascade: .

By choosing a larger value, we find that the unity gain frequency of the op-amp should not be less than 100 KHz.

· The op-amp gain must be large enough.

· The supply voltage of the op-amp must match the voltage of the power supplies, if known. Otherwise, it is advisable to select an op-amp with a wide range of supply voltages.

· When choosing an op-amp for a multi-stage high-pass filter, it is better to choose an op-amp with the lowest possible offset voltage.

According to the reference book, we will select an op-amp of type 140UD6A, structurally designed in a housing of type 301.8-2. Op amps of this type are general purpose op amps with internal frequency correction and output protection during load short circuits and have the following parameters:

Supply voltage, V

Supply voltage, V

Current consumption, mA

Offset voltage, mV

Op-amp voltage gain

Unity gain frequency, MHz1

5.4 Methodology for setting up and adjusting the developed filter

Setting up this filter is not very difficult. The parameters of the frequency response are “adjusted” using resistors of both the first and second stages independently of each other, and the adjustment of one filter parameter does not affect the values ​​of other parameters.

The setup is carried out as follows:

1. The gain is set by resistors R2 of the first and R5 of the second stage.

2. The frequency of the pole of the first stage is adjusted by resistor R1, the frequency of the pole of the second stage by resistor R4.

3. The quality factor of the second stage is regulated by resistor R8, but the quality factor of the first stage is not adjustable (constant for any element values).

The result of this course work is obtaining and calculating the circuit of a given filter. A high-pass filter with approximation of frequency characteristics by a Butterworth polynomial with the parameters given in the technical specifications is of the third order and is a two-stage connected high-pass filter of the first order (based on a non-inverting amplifier) ​​and second order (based on impedance converters). The circuit contains three operational amplifiers, eight resistors and three capacitors. This circuit uses two power supplies of 15 V each.

The choice of circuit for each stage of the general filter was carried out on the basis of the technical specifications (to ensure low sensitivity to deviations in the values ​​of the elements) taking into account the advantages and disadvantages of each type of filter circuits used as stages of the general filter.

The values ​​of the circuit elements were selected and calculated in such a way as to bring them as close as possible to the standard nominal series E24, and also to obtain the highest possible input impedance of each filter stage.

After modeling the filter circuit using the ElectronicsWorkbench5.0 package (Fig. 5.1), frequency characteristics were obtained (Fig. 5.2), having the required parameters given in the technical specifications (Fig. 2.2).

The advantages of this circuit include the ease of setting up all filter parameters, independent setting of each stage separately, and low sensitivity to deviations from the nominal values ​​of the elements.

The disadvantages are the use of three operational amplifiers in the filter circuit and, accordingly, its increased cost, as well as the relatively low input resistance (about 50 kOhm).

List of used literature

1. Zelenin A.N., Kostromitsky A.I., Bondar D.V. – Active filters on operational amplifiers. – Kh.: Teletekh, 2001. ed. second, correct. and additional – 150 pp.: ill.

2. Resistors, capacitors, transformers, chokes, switching devices REA: Reference/N.N. Akimov, E.P. Vashukov, V.A. Prokhorenko, Yu.P. Khodorenok. – Mn.: Belarus, 2004. – 591 p.: ill.

Analog integrated circuits: Reference/A.L. Bulychev, V.I. Galkin, 382 pp.: V.A. Prokhorenko. – 2nd ed., revised. and additional - Mn.: Belarus, 1993. - damn.

The frequency response of the Butterworth filter is described by the equation

Features of the Butterworth filter: nonlinear phase response; cutoff frequency independent of the number of poles; oscillatory nature of the transient response with a step input signal. As the filter order increases, the oscillatory nature increases.

Chebyshev filter

The frequency response of the Chebyshev filter is described by the equation

,

Where T n 2 (ω/ω n ) – Chebyshev polynomial n-th order.

The Chebyshev polynomial is calculated using the recurrent formula

Features of the Chebyshev filter: increased unevenness of the phase response; wave-like characteristic in the passband. The higher the coefficient of unevenness of the frequency response of the filter in the passband, the sharper the decline in the transition region at the same order. The transient oscillation of a stepped input signal is greater than that of a Butterworth filter. The quality factor of the Chebyshev filter poles is higher than that of the Butterworth filter.

Bessel filter

The frequency response of the Bessel filter is described by the equation

,

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

The Bessel polynomial is calculated using the recurrent formula

Features of the Bessel filter: fairly uniform frequency response and phase response, approximated by the Gaussian function; the phase shift of the filter is proportional to the frequency, i.e. the filter has a frequency-independent group delay time. The cutoff frequency changes as the number of filter poles changes. The filter's frequency response is usually flatter than that of Butterworth and Chebyshev. This filter is especially suitable for pulse circuits and phase-sensitive signal processing.

Cauer filter (elliptical filter)

General view of the transfer function of the Cauer filter

.

Features of the Cauer filter: uneven frequency response in the passband and stopband; the sharpest drop in frequency response of all the above filters; implements the required transfer functions with a lower filter order than when using other types of filters.

Determining the filter order

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

.

Chebyshev filter order

.

For the Bessel filter, there is no formula for calculating the order; instead, tables are provided that correspond the filter order to the minimum required deviation of the delay time from unity at a given frequency and the loss level in dB).

When calculating the Bessel filter order, the following parameters are specified:

Permissible percentage deviation of group delay time at a given frequency ω ω cp h ;

The filter gain attenuation level can be set in dB at frequency ω , normalized relative to ω cp h .

Based on these data, the required order of the Bessel filter is determined.

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

In Fig. 12.4, 12.5 show typical circuits of low-pass filter cascades.

A) b)

Rice. 12.4. Low-pass filter cascades of Butterworth, Chebyshev and Bessel: A - 1st order; b – 2nd order

A) b)

Rice. 12.5. Cauer low-pass filter cascades: A - 1st order; b – 2nd order

General view of the transfer functions of the Butterworth, Chebyshev and Bessel low-pass filters of the 1st and 2nd order

,
.

General view of the transfer functions of the Cauer low-pass filter of the 1st and 2nd order

,
.

The key difference between a 2nd order Cauer filter and a bandstop filter is that in the Cauer filter transfer function the frequency ratio Ω s ≠ 1.

Calculation method for Butterworth, Chebyshev and Bessel low-pass filters

This technique is based on the coefficients given in the tables and is valid for Butterworth, Chebyshev and Bessel filters. The method for calculating Cauer filters is given separately. The calculation of Butterworth, Chebyshev and Bessel low-pass filters begins with determining their order. For all filters, the minimum and maximum attenuation parameters and cutoff frequency are set. For Chebyshev filters, the coefficient of frequency response unevenness in the passband is additionally determined, and for Bessel filters, the group delay time is determined. Next, the transfer function of the filter is determined, which can be taken from the tables, and its 1st and 2nd order cascades are calculated, the following calculation procedure is observed:

Depending on the order and type of the filter, the circuits of its cascades are selected, while an even-order filter consists of n/2 2nd order cascades, and an odd order filter - from one 1st order cascade and ( n– 1)/2 cascades of the 2nd order;

To calculate a 1st order cascade:

The selected filter type and order determines the value b 1 1st order cascade;

By reducing the occupied area, the capacity rating is selected C and calculated R according to the formula (you can also choose R, but it is recommended to choose C, for reasons of accuracy)

;

The gain is calculated TO at U 1 1st order cascade, which is determined from the relation

,

Where TO at U– gain of the filter as a whole; TO at U 2 , …, TO at Un– gain factors of 2nd order cascades;

To realize gain TO at U 1 it is necessary to set resistors based on the following relationship

R B = R A ּ (TO at U1 –1) .

To calculate a 2nd order cascade:

By reducing the occupied area, the nominal values ​​of the containers are selected C 1 = C 2 = C;

Coefficients are selected from tables b 1 i And Q pi for 2nd order cascades;

According to a given capacitor rating C resistors are calculated R according to the formula

;

For the selected filter type, you must set the appropriate gain TO at Ui = 3 – (1/Q pi) of each 2nd order stage, by setting resistors based on the following relationship

R B = R A ּ (TO at Ui –1) ;

For Bessel filters, it is necessary to multiply the ratings of all capacitors by the required group delay time.

Institute of Non-Ferrous Metals and Gold Siberian Federal University

Department of Automation of Production Processes

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

 LPF on INUN  Biquad low-pass filters  Setting up 2nd order filters  Low-pass filter of odd order 

 Chebyshev low-pass filter II type  Elliptical low-pass filters  Elliptical low-pass filters on INUN  Elliptical low-pass filters with 3 capacitors  Biquadratic elliptical low-pass filters  Setting up the Chebyshev low-pass filter II type and elliptical 

 Setting up 2nd order filters  All-pass filters  Low-pass filter modeling  Creating diagrams 

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

Laboratory work No. 1

”Study of signal filtering in Micro-Cap 6/7 environment”

Goal of the work

1. Study the main types and characteristics of filters

2. Explore filter modeling in the Micro-Cap 6 environment.

3. Explore the characteristics of active filters in the 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 (imposition of interference signals, noise) (Fig. 1.1). There are hardware (circuit) and digital (software) filtering. In the first case, electronic filters made of passive and active elements are used, in the second case, various software methods are used to isolate and eliminate interference. Hardware filtering is used in ICD modules (communication devices with an object) of controllers and distributed data collection and control systems.

Digital filtering is used in the top-level computer control system of the automated process control system. This paper discusses in detail the issues of hardware filtering.

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

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

bandpass filters (pass a frequency band and block frequencies above and below this band);

band-stop filters (which delay a frequency band and pass frequencies above and below that band).

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

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

PF of the implemented filter has the form

Where A And b - constant values, and T , n = 1, 2, 3 ... (m £ n).

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

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

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

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

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

At low frequencies (below 0.5 MHz), the parameters of the inductors are unsatisfactory: large sizes and deviations from ideal characteristics. Inductors are poorly suited for integral design. 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 (op-amps). Op-amps must have: high gain (50 times greater than that of the filter); high rate of rise of output voltage (up to 100-1000 V/µs).

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

For first order PF filters

Where IN And WITH - constant numbers; P(s) - a polynomial of the second or lesser degree.

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

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

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

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

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

Butterworth low pass filter n-th order has the frequency response of the following form

The PF of the Butterworth filter as a polynomial filter is equal to

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

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

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

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

The PFs of the Chebyshev and Butterworth low-pass filters are identical in shape 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 former has a narrower transition region width. However, the Chebyshev filter has a worse (more nonlinear) phase response than the Butterworth filter.

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

The phase response characteristics of the Chebyshev filter for the 2nd-7th orders are shown in Fig. 1.18. For comparison, in Fig. 1.18 the dashed line shows the phase response of a 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 a high-order Chebyshev filter is better than the frequency response of a 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, a 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 permissible attenuation in the passband a 1 (dB), minimum permissible attenuation in the stopband a 2 (dB), cutoff frequency w s (rad/s) or f c (Hz) and maximum permissible transition region width T W, which is defined as follows:

where logarithms can be either natural or decimal.

Equation (1.24) can be written as

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

Similarly, based on (1.18) for K = 1 find the minimum order of the Chebyshev filter

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

Finding the nearest larger integer again, we get P= 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 considered, the Chebyshev filter provides the same slope of the transfer function as the Butterworth filter of double complexity.

1.2. LPF WITH MULTI-LOOP FEEDBACK

AND INFINITE GAIN

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

Resistances satisfying equation (1.30) are equal to

A good approach is to set the nominal value of the capacitance C 2, close to the value 10/ f c µF and select the highest available nominal capacitance value C 1 satisfying equation (1.31). The resistances should be close to the values ​​calculated by (1.31). The higher the filter order, the more critical these requirements are. If calculated nominal resistance values ​​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, assume that you want to design a second-order MOC Chebyshev filter with a ripple of 0.5 dB, a bandwidth of 1000 Hz, 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. Selecting nominal value C 2 = 10/f c= 10/1000 = 0.01 μF = 10 -8 F, from (1.32) we get

Now suppose that it is necessary to design a sixth order Butterworth filter with an MOC cutoff frequency f c= 1000 Hz and gain K= 8. It will consist of three second-order links, each with a PF 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 IN= 0.517638 and C = 1. Let us again select the nominal value of the capacitance 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 kOhm; R 1 =69.7 kOhm; R 3 = 90.9 kOhm. The other two links are calculated in a similar way, and then the links are cascaded to implement 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 and low output impedance; thus, it can be immediately cascaded with other links to implement a higher order filter. The disadvantage of the scheme 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 changes. To achieve good results, gain TO

Adjusted LPF-filter. ... MOS-structure, is the ability to adjust the gain and band filter when changing denominations minimum ... filter on microcircuits type...has the same order the same values ​​as... classic filtersChebysheva And Butterworth, ...

Plan:

Introduction

• 1 Review
  • 1.1 Normalized Butterworth polynomials
  • 1.2 Maximum smoothness
  • 1.3 High-frequency roll-off
• 2 Filter design
  • 2.1 Cauer topology
  • 2.2 Sallen-Kay topology
• 3 Comparison with other linear filters
• 4 Example
Literature

Introduction

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

Such filters were first described by British engineer Stefan Butterworth in the article “On the Theory of Filter Amplifiers.” On the Theory of Filter Amplifiers ), In the magazine Wireless Engineer in 1930.

1. Review

The frequency response of the Butterworth filter is maximally smooth at passband frequencies and decreases to almost zero at stopband frequencies. When plotting the frequency response of a Butterworth filter on a logarithmic phase response, the amplitude decreases toward minus infinity at the stopband frequencies. In the case of a first-order filter, the frequency response attenuates 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 retains the shape of the frequency response for higher orders (with the exception of a steeper roll-off of the characteristic at the suppression band), while many other types of filters (Bessel filter, Chebyshev filter, elliptic filter) have different shapes of the frequency response at different orders.

Compared to Chebyshev types I and II filters or the elliptical filter, the Butterworth filter has a flatter rolloff and therefore must be of higher order (which is more difficult to implement) in order to provide the desired performance at stopband frequencies. However, the Butterworth filter has a more linear phase-frequency response at passband frequencies.

The frequency response for low-pass Butterworth filters is of the order of 1 to 5. The slope of the characteristic is 20 n dB/decade, where n- filter order.

As with all filters, when considering frequency characteristics, a low-pass filter is used, from which you can easily obtain a high-pass filter, and by connecting several such filters in series, a band-pass filter or notch filter.

The frequency response of a 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 passed with a gain, and frequencies above the cutoff frequency will be completely suppressed. For finite values, the decline in the characteristic will be gentle.

Using a formal replacement, we present the expression as:

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 a Butterworth filter can only be determined 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 applies to digital Butterworth filters, with the only difference being that the relationships are not written for s-plane, and for z-plane.

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 relations with real coefficients (complex conjugate pairs are combined using multiplication). The polynomials are normalized by the cutoff frequency: . The normalized Butterworth polynomials thus have the following canonical form:

Below are the Butterworth polynomial coefficients for the first eight orders:

1.2. Maximum smoothness

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

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

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

1.3. High-frequency roll-off

Having accepted, we find the slope of the logarithm of the frequency response 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 with which linear analog filters are implemented. These schemes differ only in the values ​​of the elements, but the structure remains unchanged.

2.1. Cauer topology

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

2.2. Sallen-Kay topology

The Sallen-Kay topology uses, in addition to passive ones, also active elements (operational amplifiers and capacitors). Each stage of the Sallen-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 valid pole is found, it must be implemented separately, usually as an RC circuit, and included in the overall circuit.

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

The denominator must 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 linear filters

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

It can be seen from the figure that the Butterworth filter roll-off is the slowest of the four, but it also has the smoothest frequency response at passband frequencies.

4. Example

Analog low-pass Butterworth filter (Cauer topology) with cutoff frequency with the following element values: farad, ohm, and henry.

Logarithmic density plot of the transfer function H(s) on the complex argument plane for a third-order Butterworth filter with cutoff frequency . The 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. Indicating the total resistance of the capacitors C How 1/Cs and impedance of inductances L How Ls, where is a complex variable, and using 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 response is given by the equation:

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

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

By replacing each inductance with a capacitance, and the capacitances with inductances, we obtain a high-pass Butterworth filter.

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

Literature

• V.A. Lucas Theory of automatic control. - M.: Nedra, 1990.
• B.H. 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. - 4th 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

In filters, the calculation usually begins with setting the filter parameters, the most important of which is the frequency response. As we have already discussed in the article, first the requirements of a given filter are brought to the requirements of the low-pass filter prototype. An example of the requirements for the amplitude-frequency response of a low-pass filter prototype of the designed filter is shown in Figure 1.

This graph shows the dependence of the filter transmission coefficient on the normalized frequency ξ , Where ξ = f/f V

The graph shown in Figure 1 shows that the permissible unevenness of the transmission coefficient is specified in the passband. In the stopband, the minimum coefficient of suppression of the interfering signal is set. The real filter can have any shape. The main thing is that it does not cross the boundaries of the specified requirements.

For quite a long time, the filter was calculated by selecting the amplitude-frequency response using standard links (m-link or k-link). This method was called the application method. It was quite complicated and did not provide the optimal ratio of the quality of the developed filter and the number of links. Therefore, mathematical methods have been developed for approximating the amplitude-frequency response with given characteristics.

In mathematics, approximation is the representation of a complex relationship by some known function. Usually this function is quite simple. When developing a filter, it is important that the approximating function can be easily implemented in circuitry. To do this, the functions are implemented using the zeros and poles of the transmission coefficient of a four-port network, in this case a filter. They are easily implemented using LC circuits or feedback loops.

The most common type of approximation of the frequency response of a filter is the Butterworth approximation. Such filters are called Butterworth filters.

Butterworth filters

A distinctive feature of the amplitude-frequency response of the Butterworth filter is the absence of minima and maxima in the passband and delay band. The frequency response rolloff at the edge of the passband of these filters is 3 dB. If a filter is required to have a lower ripple value in the passband, then the correct filter frequency f in is selected above the specified upper frequency of the passband. The frequency response approximation function for the low-pass filter prototype of the Butterworth filter is as follows:

Where ξ — normalized frequency;
n— filter order.

In this case, the real amplitude-frequency characteristic of the filter being developed can be obtained by multiplying the normalized frequency ξ to the filter cutoff frequency. For a low-pass Butterworth filter, the frequency response approximation function will look like this:

Now let us note that when calculating filters, the concept of a complex s-plane is widely used, on which the circular frequency is plotted along the ordinate axis jω, and along the x-axis is the reciprocal of the quality factor. In this way, it is possible to determine the main parameters of the LC circuits that are part of the filter circuit: tuning frequency (resonant frequency) and quality factor. The transition to the s-plane is carried out using .

A detailed derivation of the pole positions of the Butterworth filter on the complex s-plane is given in. The main thing for us is that the poles of this filter are located on the unit circle at an equal distance from each other. The number of poles is determined by the order of the filter.

Figure 2 shows the pole locations for a first order Butterworth filter. The frequency response corresponding to a given arrangement of poles on the complex s-plane is shown nearby.

Figure 2 shows that for a first-order filter, the pole must be tuned to zero frequency and its quality factor must be equal to unity. The frequency response graph shows that the tuning frequency of the pole is indeed zero, and the quality factor of the pole is such that at the cutoff frequency of the normalized Butterworth filter, equal to unity, its transmission coefficient is −3 dB.

The poles for the second order Butterworth filter are determined in exactly the same way. This time, the pole tuning frequency is selected at the intersection of the unit circle with a straight line passing through the center of the circle at an angle of 45°. An example of the location of the poles on the complex s-plane and the frequency response of a second-order Butterworth filter is shown in Figure 3.

In this case, the resonant frequency of the pole is located close to the cutoff frequency of the normalized filter. It is equal to 0.707. The pole quality factor according to the pole location graph is the root of two times higher than the pole quality factor of a first-order Butterworth filter, so the slope of the amplitude-frequency response is greater. (Pay attention to the numbers on the right side of the graph. With a frequency detuning of 2, the suppression is already 13 dB) The left side of the amplitude-frequency response of the pole turns out to be flat. This is due to the influence of the pole located in the negative frequency zone.

The location of the poles and the amplitude-frequency response of the third-order Butterworth filter is shown in Figure 4.

As can be seen from the graphs shown in Figures 2...5, as the order of the Butterworth filter increases, the slope of the amplitude-frequency response increases and the required quality factor of the second-order circuit (circuit) that implements the pole of the filter’s transmission characteristic increases. It is the increase in the required quality factor that limits the maximum order of the filter that can be implemented. Currently, it is possible to implement Butterworth filters up to the eighth - tenth order.

Chebyshev filters

In Chebyshev filters, the amplitude-frequency response is approximated as follows:

In this case, the amplitude-frequency response of a real Chebyshev filter, just like in the Butterworth filter, can be obtained by multiplying the normalized frequency ξ to the cutoff frequency of the filter being developed. For a low-pass Chebyshev filter, the amplitude-frequency response can be determined as follows:

The amplitude-frequency response of the low-pass Chebyshev filter is characterized by a steeper decline in the frequency range above the upper pass frequency. This gain is achieved due to the appearance of frequency response unevenness in the passband. The unevenness of the approximation function of the frequency response of the Chebyshev filter is caused by the higher quality factor of the poles.

A detailed derivation of the position of the poles of the approximating function of the Chebyshev filter on the s-plane is given in. What is important for us is that the poles of the Chebyshev filter are located on an ellipse, the major axis of which coincides with the axis of normalized frequencies. On this axis, the ellipse passes through the cutoff frequency point of the low-pass filter.

In the normalized version, this point is equal to one. The second axis is determined by the unevenness of the frequency response approximation function in the passband. The greater the permissible ripple in the passband, the smaller this axis. There is a kind of “flattening” of the unit circle of the Butterworth filter. The poles seem to be approaching the frequency axis. This corresponds to an increase in the quality factor of the filter poles. The greater the unevenness in the passband, the greater the quality factor of the poles, the greater the rate of increase in attenuation in the stopband of the Chebyshev filter. The number of poles of the frequency response approximation function is determined by the order of the Chebyshev filter.

It should be noted that there is no first order Chebyshev filter. The location of the poles and frequency response of the second-order Chebyshev filter is shown in Figure 5. The characteristic of the Chebyshev filter is interesting in that the frequencies of the poles are clearly visible on it. They correspond to the maximum frequency response in the passband. For a second-order filter, the pole frequency corresponds to ξ =0.707.