Active frequency-selective filters designing

Аuthors

Burenko E. A.^*, Nesterov S. V.^**

,

*e-mail: super.evgeny-burenko2012@yandex.ru
**e-mail: nesterov.vzlet@mail.ru

Abstract

While their studying, students of radio engineering specialties necessarily face the task of designing electric frequency-selective filters [1, 2, 3]. Due to the trend towards microminiaturization and difficulties in constructing small-sized filtering devices with inductors for radio engineering systems, linear active RC-filters 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 four-pole 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 (four-pole). 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 first-order low-pass filter with a transfer function in the form of (2) can be implemented, if a parallel RC-circuit is used in the feedback circuit of the operational amplifier.

It is quite simple to implement a second-order 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 low-pass active RC-filter of the fifth order was developed (Fig. 1), realized with the characteristic of the Legendre polynomials (Fig. 2).

Similarly, the scheme of the active low-pass RC-filter 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 high-pass 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 high-pass filter of the seventh order with amplitude-frequency response on the Legendre polynomials is implemented based of the known first- and second order links.

The scheme of the active high-pass filter of the seventh order with an optimum monotone characteristic was also realized.

By replacing variables, one can convert the amplitude-frequency responce of the low pass filter to the amplitude-frequency 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 cut-off 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 amplitude-frequency 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.

High-order cut-off 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 RC-circuits, amplitude-frequency response

References

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