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Mismatched ladder filters

The features of oscillation pattern in mismatched finite electric ladder filters

E-mail: selftrans@yandex.ru , selflab@mail.ru

Basing on the original relationship of the Dynamical ElectroMechanical Analogy DEMA and original exact analytical solutions for a lumped mechanical elastic line as an analogue, it is studied, how the load resistance effects on the amplitude-frequency and phase-frequency characteristics of mismatched finite ladder filters. It is shown that in filters of such type the indicated characteristics have a brightly expressed resonance form and essentially transform in the lower and medial domains of the pass band, changing insufficiently in the vicinity of cutoff frequency. It disables the conventional method to determine the total phase delay and the ladder filter transmission coefficient and requires finding the exact analytical solutions by way of presented method. The obtained calculated regularities well agree with the experimental results for similar-parameters ladder filters. The obtained results can be extended to essentially more complicated ladder-filter circuits.

Keywords: Electric ladder filters; electromechanical analogy; elastic lumped lines; ODE

Classification by MSC 2000: 30E25; 93A30; 93C05; 94C05

Classification by PASC 2001: 02.60.Li; 84.30.Vn; 84.40.Az

1. Introduction

The basic calculation method for electric ladder filters is the two-port method and those developed on its basis. It is accepted that “one of the two-port method advantages is that the complicated circuit can be reduced to a few two-ports connections. The coefficients of each are easy expressed through the element parameters of the related sections of the circuit. And a few two-ports connection in its turn can be presented as some resulting two-port. The finding of the coefficient matrix is reduced mostly to summing and multiplying the matrixes of the separate two-ports into which the considered circuit is factored” [1, p. 53]. With the obviously simple and effective approach, this method has essential restrictions when applied. Specifically, when calculating ladder filters, it is supposed that the input and output of their sections are matched, as “usually one seeks to insert the separate sections of the laddered circuit matched” [2, p. 269]. The more, “the inserted sections must be matched always, as only at this condition one can sum the characteristic constants of the transmission” [3, p. 120]. Just because of it “if the circuit was set up of T-sections, it must begin and finish with the impedance ₁/2, and if of pi-sections – with the parallel impedance 2₂” [4, p. 603]. "Considering the filter circuits, one supposes that the filter input and output are matched with the source of e.m.f. and with the load, i.e., that each section of the filter and the circuit on the whole are loaded on the impedance equal to that characteristic. In the reality this condition is not satisfied, as the characteristic impedance of the filter depends on frequency; it has a real value at the transparency band and is reactive at the stop band” [4, p. 623]. Furthermore, “unfortunately, the characteristic impedances of sections are expressed by the functions physically unrealisable, and by this reason one can proceed the matched connection in real circuits only approximately” [3, p. 120].

This misfit of the two-port method approach to the real processes in ladder filters is caused by the fact that under the mismatched load the influence of the wave properties of the filter as a whole is revealed. For example, in [5] in case of many-sectioned wave-guides, it is experimentally established that “the lengths of the arbitrarily chosen sections scatter coherently the chopped reflections and make the ripples … The reflected signal amplitude growing in excess of its end-to-end response can be raised by single reflected signals from the points far from the source” [5].