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 Getting the job done Security questions
Laboratory work No. 1
”Study of signal filtering in Micro-Cap 6/7 environment”
Purpose of the work
1. Study the main types and characteristics of filters
2. Explore filter modeling in the Micro-Cap 6 environment.
3. Research characteristics active filters in Micro-Cap 6 environment
Theoretical information
1. Types and characteristics of filters
Signal filtering plays an important role in digital systems management. 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.
The following types of filters are distinguished:low-pass filters - low-pass filters (pass low frequencies and delay high frequencies);
filters treble(pass high frequencies and delay 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.
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 - 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:
|
Rice. 1.8. LPF ( A) and its frequency response ( b) |
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).
|
Rice. 1.9. T- and U-shaped low-pass filters |
Even order filter with n > 2 contains n/2 second-order links connected in cascade. Odd order filter with n > 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 its value transfer function at s = 0 or the value of its frequency response at w = 0 , i.e. . equals A.
The following types of low-pass filters are distinguished: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 n- Chebyshev polynomial of the first kind of degree n, 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.
Frequency response of the Chebyshev filter of this order better than the Butterworth frequency response, 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 general characteristics the maximum permissible attenuation in the passband is specified 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
w с /w 1 = ( T W/w c) + 1 |
and substitute the resulting relation into (1.25) to find the order dependence n 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 n= 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
|
Rice. 1.11. Low-pass filter with second-order MOS |
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.
This scheme implements equation (1.29) with inverting gain – TO(TO> 0) andResistances 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 using (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.
Due to its relative simplicity, the MOC filter is one of the most popular types filters with inverting gain. 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, ...
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Let us determine the order of the filter based on the required conditions according to the graph for attenuation in the stopband in the book by G. Lamb “Analog and digital filters» Chapter 8.1 p.215.
It is clear that a 4th order filter is sufficient for the required attenuation. The graph is shown for the case when w c = 1 rad/s, and, accordingly, the frequency at which the necessary attenuation is needed is 2 rad/s (4 and 8 kHz, respectively). General graph for the transfer function of a Butterworth filter:
We define the circuit implementation of the filter:
active fourth-order low-pass filter with complex negative feedback:
In order for the desired circuit to have the desired amplitude-frequency response, the elements included in it can be selected with not very high accuracy, which is an advantage of this circuit.
Fourth order active low pass filter with positive feedback:
In this circuit, the gain of the operational amplifier must have a strictly defined value, and the transmission coefficient of this circuit will not be more than 3. Therefore this diagram can be discarded.
Fourth order active low pass filter with ohmic negative feedback
This filter is built on four op-amps, which increases noise and the complexity of calculating this circuit, so we also discard it.
From the considered circuits, we select a filter with complex negative feedback.
Filter calculation
Definition of transfer function
Recording table values coefficients for the fourth order Butterworth filter:
a 1 =1.8478 b 1 =1
a 2 =0.7654 b 2 =1
(see U. Titze, K. Schenk “Semiconductor circuitry” table 13.6 p. 195)
The general expression of the transfer function for a fourth-order low-pass filter is:
(see U. Titze, K. Schenk “Semiconductor circuitry” table 13.2 p. 190 and form 13.4 p. 186).
The transfer function of the first link has the form:
The transfer function of the second link has the form:
where w c is the circular cutoff frequency of the filter, w c =2pf c .
Calculation of part ratings
Equating the coefficients of expressions (2) and (3) to the coefficients of expression (1), we obtain:
Transfer coefficients constant signal for cascades, their product A 0 should be equal to 10 as specified. They are negative, since these stages are inverting, but their product gives a positive transmission coefficient.
To calculate the circuit, it is better to specify the capacitances of the capacitors, and in order for the value of R 2 to be valid, the condition must be met
and accordingly 
Based on these conditions, C 1 = C 3 = 1 nF, C 2 = 10 nF, C 4 = 33 nF are selected.
We calculate the resistance values for the first stage:
Resistance values of the second stage:
Op amp selection
When choosing an op-amp, it is necessary to take into account the frequency range of the filter: the unity gain frequency of the op-amp (at which the gain is equal to unity) must be greater than the product of the cutoff frequency and the filter gain K y.
Since the maximum gain is 3.33 and the cutoff frequency is 4 kHz, almost all existing op-amps satisfy this condition.
Another important parameter of an op-amp is its input impedance. It should be greater than ten times the maximum resistance of the circuit resistor.
The maximum resistance in the circuit is 99.6 kOhm, therefore the input resistance of the op-amp must be at least 996 kOhm.
It is also necessary to take into account the load capacity of the op-amp. For modern op-amps, the minimum load resistance is 2 kOhm. Considering that the resistances R1 and R4 are equal to 33.2 and 3.09 kOhms, respectively, the output current of the operational amplifier will certainly be less than the maximum permissible.
In accordance with the above requirements, we select the K140UD601 OU with the following passport data (characteristics):
K y. min = 50,000
Rin = 1 MOhm
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 specifications
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 telecommunications could not but cause feelings of dissatisfaction. Digital nodes implemented with widespread use integrated circuits(IC), stood out for 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 (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 made high-quality linear integrated circuits, including integrated operational amplifiers (OPA) available to developers. These op-amps, having large functionality, 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. In the future 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. Job digital filters is provided mainly software, so they are much more flexible in use compared to analogue 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) - transmit 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 nominal values of the elements 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 rectangular 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 links is carried out if a filter order higher than the second is needed, that is, when it is necessary to form 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
According to the conditions of the task, we find the ones we need boundary conditions filter frequencies. 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. in the form
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 because general order of serially connected active filters 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 the 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 with 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 E24 series, 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 frequency response unevenness coefficient 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 order 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 is 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.
Lesson topic 28: Classification of electrical filters.
28.1 Definitions.
An electric frequency filter is a four-port network that passes currents of some frequencies well with low attenuation (3 dB attenuation), and currents of other frequencies poorly with high attenuation (30 dB).
The range of frequencies in which there is little attenuation is called the passband.
The range of frequencies in which the attenuation is large is called the stopband.
A transition strip is introduced between these stripes.
The main characteristic of electric filters is the dependence of the operating attenuation on frequency.
This characteristic is called the frequency attenuation characteristic.
- cutoff frequency at which the operating attenuation is 3 dB.
- permissible attenuation, set by the mechanical parameters of the filter.
- permissible frequency corresponding to permissible attenuation.
PP passband – the frequency range in which 
dB.
PB - stopband - frequency range in which the operating attenuation is greater than permissible.
28.2 Classification
1
By bandwidth location:
a) LPF – low pass filter – passes low frequencies and blocks high ones.
It is used in communication equipment (TV receivers).
b 
) HPF - high pass filter - passes high frequencies and delays low ones.
V 
) PF - bandpass filters - pass only a certain frequency band.
G 
) SF - notch or blocking filters - do not pass only a certain frequency band, and let the rest pass.
2 According to the element base:
a) LC filters (passive)
b) RC filters (passive)
c) active ARC filters
d) special types of filters:
Piezoelectric
Magnetostrictive
3 For mathematical support:
A 
) Butterworth filters. Operating attenuation characteristic 
has a value of 0 at 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 a “waviness” in the passband, which is a disadvantage.
c) Zolotarev filters. Operating attenuation characteristic 
in the passband it has undulations, and in the stopband there is a dip in characteristics.
Lesson topic 29: Low-pass and high-pass Butterworth filters.
29.1 Butterworth LF.
Butterworth proposed the following attenuation formula:
,dB
Where 
- Butterworth function (normalized frequency)
n – filter order
For low pass filter 
, Where 
- any desired frequency
- cutoff frequency, which is equal to 
To implement this characteristic, L and C filters are used.
AND 
The inductance is placed in series with the load, since 
and with growth 
increases 
Therefore, low-frequency currents will easily pass through the inductance resistance, and high-frequency currents will be delayed and will not reach the load.
The capacitor is placed in parallel with the load, since 
, therefore the capacitor passes high-frequency currents well and poorly lower ones. High frequency currents will be closed through the capacitor, and low frequency currents will pass to the load.
The filter circuit consists of alternating L and C.
Butterworth low-pass filter 3rd order T-shaped
Butterworth low-pass filter. 3rd order U-shaped.
Connection
