V.2 No 1

39

Mismatched ladder filters

Furthermore, in the relationship (8) the external excitation frequency is important, because the initial system (1) modelling a mechanical line is the system of the second-order ODE with constant coefficients, while an electrical line is modelled by the integro-differential equations (5). This is why, before establishing the relationship between the models, one had to reduce both systems to a common algebraic form. The dependence on frequency appearing in the relationship (8) became the direct consequence of this reduction. And one cannot yield the relationship (8) by the differentiation or integration within the frames of conventional approach. Only transiting from the system of differential equations to the algebraic system, the above feature in the relationship of the electromechanical analogy appears.

The basic difference between the conventional analogy systems in the Table 1 and the system (8) furnishes to say that the ladder filters are inherent in their own system of correspondence which due to its features has been determined as the Dynamical System of ElectroMechanical Analogy (DEMA).

One of its main advantages is that it is based on the complete complex of solutions for a mechanical elastic line, including the solutions for forced and free vibrations. Connecting the different fields of knowledge with the help of DEMA, one can, using the new results in mechanics, first, to provide the development in the field of ladder filters (and with them the cascades, networks, transmission lines etc,); second, to combine the diversiform models, providing the wider scope for the mathematical and physical modelling of the most diversiform by their nature processes and systems. In this way we approach to the most general concept of the analogous models having been formulated by Karplus: “Two systems are analogous if their reactions to the similar excitations reveal in a similar form” [8, p. 36].

Fig. 2. The schematic diagram of a finite mechanical elastic line with unfixed end (a) and of corresponding ladder filter (b)

3. The unloaded and shorted finite ladder filters

Using the DEMA relationship and results [11], determine the characteristics of an unloaded and shorted ladder filter. In Fig. 2 the diagrams of mechanical elastic line with unfixed end containing n masses are shown as well as an equivalent ladder filter with an open output. On the basis of solutions presented in [11] and relationship (8), the solutions for an unloaded electrical filter will have the following form:

at the pass band, _el <1

at the stop band, _el >1

and at the cutoff frequency, _el =1

where is the external current I (t) frequency,  is the initial phase of an external current I (t) ,  _el = (-₁ / 4₂ )^1/2 , _el = arcsin _el ,  _el+ = _el + (_el - 1)^1/2 ,  _el- = _el - (_el - 1)^1/2 , i = 1, 2, ... , n  is the number of the studied node of the filter, and n in this case and further is the number of the calculated nodes of the filter.

As conventionally, at the pass band the solutions describe the standing vibrations with n resonances arising at the condition