Analog aktif tüm-geçiren filtrelerin gerçeklenmesi

Abstract

Tüm-geçiren filtreler özellikle gecikme elemanı ya da faz eşitleyici (phase equalizer) olarak elektronik mühendisliğinde geniş bir kullanım alanına sahiptirler. Bu yüzden, tüm-geçiren filtreler ile ilgili gerek aktif gerekse pasif devreler için pek çok sentez ve tasarım yöntemi geliştirilmiştir ve optimum tasarım koşullarını elde edebilmek için araştırmalar sürdürülmektedir. Bu çalışmada, tüm-geçiren filtreler ile bazı makalelerden ve lineer sistemlerin kararlılığını inceleyen bir test prosedüründen hareketle, analog tüm-geçiren filtrelerin gerçeklenmesi ile ilgili bir yöntem önerilmiş ve buna bağlı olarak m 'inci dereceden bir tüm- geçiren fonksiyonun aktif-RC ve OTA-RC devreleri ile tasarımı konusunda bir yöntem geliştirilmiştir. Değişik dereceli fonksiyonların farklı devreler ile tasarım örnekleri verilmiş, ayrıca aynı fonksiyonun aynı yapıda fakat farklı sayıda eleman ile de gerçeklenebileceği gösterilmiştir. Devre elemanlarına bağlı duyarlıklar incelenmiş ve olumlu sonuçlar elde edildiği ortaya konmuştur. Ayrıca, iki ayrı tüm-geçiren fonksiyonun toplam ve farkından ikinci dereceden band-geçiren ve band-söndüren filtre fonksiyonlarının sağlandığı gösterilmiştir.

Filters, whether theoretically or practically, have been a great and significant part of electronics engineering for several decades. Because they can take part almost in every electronic circuit, they also constitute one of the mostly researched areas. All -pass filters form a sub-section of this area. In contini- ous-time domain, they can be represented with the transfer function Am(s) ("A" for all-pass and "m" for the order of the function) which has a constant magnitude for all m, namely unity. Properties of Am(s) are as follows: T. A(s)«İiL U^)= -51W 'D(s) m D(» 2. Am(s).Am(-s)=l, for all s. 3. If A (s) has a zero at s=a0, then it has a pole at s=-a0. 4. If Am(s) has a zero at s=a-]+jbi, then it has a pole at s=-a-|+jbi 4=^. the poles and zeros of Am(s) are negative complex conjugate. Because all-pass filters provide constant magnitude of unity, they can be used as delay elements or phase-shifters. Like other types of filters, all -pass filters can be realized with active or passive components in continous- or discrete-time domain. Studies to obtain optimum conditions for filter design are carried on and new methods are brought out. Today, we have a large choice for realization. It has been achieved through different research operations and development stages; Going back to early 70 's, a new method for realizing active all-pass filters using Wien's bridge [15] can be taken into consi deration. The circuit that was proposed for realization combinedthe structure of Mien's bridge with operational amplifiers and different versions of RLC circuits. Another design method used a single transistor for active ele ments. It was based on the connection of series RLC circuits bet ween collector-base junction of the common-emitter circuit consist ing of the:single transistor. The number of parallel connections of series RLC circuits denoted the order of the function (for further information, the reader is referned to [10j.). Together with the progress in integrated circuit technology, the use of inductive elements in filter design became more and more inconvenient because those elements were not suitable for integratr ion and for low-frequency circuits. Therefore, active-RC and active- C circuits which could easily be produced applying this technology replaced them and proved themselves very suitable for filter design. Like for other types of filters, active-RC and active-C filters are among the most popular ones in analog all-pass filtering. Many Researches have been carried out on subjects like dynamic performance, sensitivity performance and obtaining a broad frequency range. These studies are carried out today, too. A simulation of the inductor resulted in obtaining a synthesis method using only capacitors and gyrators as active elements. Another active element used in all-pass filter circuits is the current conveyor [8], [12], Besides, there are also circuits that use dependent sources like voltage-controlled voltage source or voltage-controlled current source as active elements. Their advant age to other active elements (especially to operational amplifiers) is that they can be controlled via an external signal (which is not possible for operational amplifiers). The tendency to produce analog active filters as a single chip led scientists and engineers to elements which were suitable for that technology. As a result of this, OTAs (operational transcon- ductance amplifier) started to be used more commonly in active filter design procedures. This preferability comes from the good perfor mance of the OTA at high frequencies and linear controllability of its transconductance gain via an external signal. OTA element can easily be applied' to MOS-technology, especially to CMOS structures, and that increases its significance in filter design. An earthed capacitor forms a suitable element pair -w^'th OTA because of its convenience in integrated circuit technology. Using OTA elements, it is possible to remove undesirable effects in the filter charac teristics by controlling its transconductance gain. In this study, the design procedure is carried out with active- RC circuits and with OTA-earthed RC circuits which can easily be turned to OTA-earthed capacitor equivalents by replacing resistors viwith OTA elements. This work presents a method for realizing analog active all- pass filters, obtaining an analog of an algorithm to realize digital transfer functions. The method is based on the realization with lossless two-pairs [14] in a cascaded lattice form. The circuit realized with this method consists of basic cells connected in cas cade whereas every element has the same value with the exception of a few elements which have to be calculated separately for every basic cell. The main idea of the circuit is to obtain another all-pass function of one order less in every basic cell. The reduction process goes up to zero-th order. The all-pass function of zero-th order is 1 (or -1 according to the sign of the given transfer funct ion). Therefore, the output terminals of the last lattice cell representing the first order are connected together. As a result, a given all-pass transfer function of m-tlr order is realized with m lattice cells connected in cascade and the output terminals of the last cell connected together. So, the input and the output variables are the input and the output voltages at the input termi nal of the first lattice cell. The way to obtain a transfer function of one order less results in finding out a coefficient km which can be achieved by taking s=am in the given m-th order transfer function ^(s).am can be any positive real constant. The multiplication of a reversed all-pass function of first order with the difference of G^s) and km divided by one minus Gm(s) times km gives another all -pass function of m-1- st order. This process is carried out calculating km_-|, km_2'--etc. until the transfer function is found as unity (For further details, the reader is referred to [4], [7] and D^]-)- The design process has been carried out both for op. amp.-RC circuits and for OTA-RC circuits. It resulted in six different prototypes of the main lattice cell from which only two-one of op. amp.-RC building blocks, another of OTA-RC building blocks-have a practical significance. Because of the order-reduction process and the connections in cascade this method requires a great number of active and passive, elements in total which may seem as a problem. But in fact, the advantages of the method are supposed to overcome its drawbacks. For ^example, the circuits realized with this method have a good sensitivity performance. Especially, the sensitivity of Gm(s) with respect to km and am comes out to be purely imaginary. This means" that choosing any real positive number as am and thus finding any constant as km does not affect the magnitude of the given function. Besides, by choosing am equal to a positive real zero of the function-if there exists one-it is possible to eliminate a vnpart of the lattice cell and so, to decrease the number of the elements realizing the circuit. The chosen OTA-RC structure is more convenient from practical point of view though the other op.amp.-RC structure realizes the circuit with less elements. The reason for this is» that OTA's are more suitable for integration and that their performance at higher frequencies are better than operational amplifiers. It is also possible to replace resistors with their OTA-equivalents so that one can achieve OTA-earthed capacitor circuits which is prefer red in integrated circuit technology. Besides, this method provides a simple topology of the circuits. This facilitates their mass- production. Another possible drawback which may appear while applying the method is that the coefficients k-j happen to be negative, this problem may arise as a result of the arrangement of the coefficients in the all-pass transfer function. One solution may be found by increasing the values given to a.j 's and by controlling whether there exists a positive value of the coefficients k-j. If there is none, one çan try to realize negative resistors (by using negative-impe dance converter etc.). Concerning second-order functions, pole frequency -Up and quality factor Qp are functions of k] and k2 so that it is possible to change their values by changing k-, and k2> The sum of unity with a second-order all-pass function gives a band-reject filter of se cond order and the difference of unity and a second-order all-pass filter provides a band-pass filter of second order.