Active frequencyselective filters designing
Radio engineering, including TV systems and devices
Аuthors
^{*}, ^{**}*email: super.evgenyburenko2012@yandex.ru
**email: nesterov.vzlet@mail.ru
Abstract
While their studying, students of radio engineering specialties necessarily face the task of designing electric frequencyselective filters [1, 2, 3]. Due to the trend towards microminiaturization and difficulties in constructing smallsized filtering devices with inductors for radio engineering systems, linear active RCfilters are widespread [4]. The article presents the examples of filters synthesis for various purposes.
The task of the filter synthesis consists of several stages:
– approximating function selection for a specified filter characteristic;
– the filter transfer function constructing according to the approximating function (any characteristic of the filter as a fourpole can be reduced to the transfer function and vice versa);
– determining the filter structure corresponding to the obtained transfer function;
– filter design and analysis of its characteristics.
There are many ways to construct a filter with a given transfer function of the desired order. One of them consists in presenting the transfer function H(p) as a product of cofactors H1(p), H2(p),.... HN(p) of the transfer functions of the first and second orders, with each cofactor is realized by a separate link (fourpole). Further, these links are connected to each other in cascade, i.e. the output of the first link is the input of the second.
A firstorder lowpass filter with a transfer function in the form of (2) can be implemented, if a parallel RCcircuit is used in the feedback circuit of the operational amplifier.
It is quite simple to implement a secondorder filter. To implement the low pass filters, highpass and bandpass filters, the second order filters schemes found wide application.
Based on the above said links, the scheme of the lowpass active RCfilter of the fifth order was developed (Fig. 1), realized with the characteristic of the Legendre polynomials (Fig. 2).
Similarly, the scheme of the active lowpass RCfilter of the sixth order is shown on Fig. 3 and its optimal monotonic response is shown on Fig. 4.
To obtain the transfer function of the highpass filter, it is enough in to replace the p operator with 1/p in equation (1):
where K_{∞} – the gain of the upper frequencies of the filter.
The scheme of the active highpass filter of the seventh order with amplitudefrequency response on the Legendre polynomials is implemented based of the known first and second order links.
The scheme of the active highpass filter of the seventh order with an optimum monotone characteristic was also realized.
By replacing variables, one can convert the amplitudefrequency responce of the low pass filter to the amplitudefrequency responce of the band pass filter. To do this, the following variables must be replaced in the transfer function (1):
where is the normalized bandwidth of a bandpass filter, defined as
where f_{–}_{x} и f_{x} are respectively the lower and upper cutoff frequencies of a passband of a bandpass filter.
Based on the first and second the order links the schemes of bandpass filters of the fifth order with amplitudefrequency responce on polynomials of Legendre and with optimum monotone approximation of amplitude –frequency characteristic are realized.
The transfer function of the cutoff filter can be obtained fr om the transfer function of the low pass filter (1) by a frequency domain conversion:
where is the normalized bandwidth of the bandstop filter.
Highorder cutoff filters can be constructed by cascading the considered links, parameters of the elements and characteristics are determined by the selected approximation.
Keywords:
filters synthesis, active RCcircuits, amplitudefrequency responseReferences

Gusev V.Yu., Krapivenko A.V. Trudy MAI, 2012, no. 50, available at: http://trudymai.ru/eng/published.php?ID=28805

Martirosov V.E., Alekseev G.A. Trudy MAI, 2013, no. 71, available at: http://trudymai.ru/eng/published.php?ID=47082

Tomozov D.A. Trudy MAI, 2011, no. 45, available at: http://trudymai.ru/eng/published.php?ID=25518&PAGEN_2=2

Tikhomirov A.V., Omel’yanchuk E.V., Semenova A.Yu. Trudy MAI, 2016, no. 91, available at: http://trudymai.ru/eng/published.php?ID=75644

Krikov D.S. Trudy MAI, 2018, no. 98, available at: http://trudymai.ru/eng/published.php?ID=90403

Lobachev N.Yu. Trudy MAI, 2017, no. 92, available at: http://trudymai.ru/eng/published.php?ID=77156

Trofimov D.V., Baev A.B. Trudy MAI, 2015, no. 84, available at: http://trudymai.ru/eng/published.php?ID=63100

Khromtsev A.V., Fomin A.I. Trudy MAI, 2008, no. 30, available at: http://trudymai.ru/eng/published.php?ID=7524

Ulakhovich D.A. Osnovy teorii lineinykh elektricheskikh tsepei (Fundamentals of the theory of linear electrical circuits), SaintPetersburg, BKhVPeterburg, 2009, 816 p.

Nesterov S.V. Sintez lineinykh elektricheskikh tsepei (Synthesis of linear electric circuits), Volgograd, Izdvo VolGU, 2013, 84 p.

Moshits G., Khorn P. Proektirovanie aktivnykh fil’trov (Active filters designing), Moscow, Mir, 1984, 320 p.

Filinyuk N.A. Aktivnye SVCh fil’try na tranzistorakh (Transistor active microwave filters), Moscow, Radio i svyaz’, 1987, 112 p.

Baskakov S.I. Radiotekhnicheskie tsepi i signaly (Electronic circuits and signals), Moscow, Vysshaya shkola, 2016, 528 p.

Chryssomallis M. T., Sahalos J.N. Filter synthesis using products of Legendre polynomials, Electrical Engineering, 1999, vol. 81, no. 6, pp. 419 – 424,

Živaljević D., Stamenković N., Stojanović V. Nearly monotonic passband lowpass filter design by using sumofsquared Legendre polynomials, International Journal of 10 Circuit Theory and Applications, 2016, vol. 44, no. 1, pp. 147 – 161, аvailable at: http://dx.doi.org/10.1002/cta.2068.

Gusev V.G., Gusev Yu.M. Elektronika (Electronics), Moscow, Vysshaya shkola, 1991, 622 p.

Volovich G.I. Skhemotekhnika analogovykh i analogotsifrovykh elektronnykh ustroistv (Analog and analogdigital electronic devices’ circuit design), Moscow, Izdatel’skii dom “DodekaXXI”, 2005, 528 p.

Opadchii Yu.F. Analogovaya i tsifrovaya elektronika (Analog and digital electronics), Moscow, Goryachaya liniya–Telekom, 2005, 768 p.

Karlashchuk V.I. Elektronnaya laboratoriya na IBM PC. Programma Electronics Workbench i ee primenenie (Electronic laboratory on IBM PC. Electronics Workbench software and its application). “SalonR”, Moscow, SalonR, 2000, 506 p.

Prikota A., Sorokin S. Sovremennaya elektronika, 2015, no. 9, pp. 62 – 65.
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