V.2 No 1 |
39 |
Mismatched ladder filters |
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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].
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Fig. 2. The schematic diagram of a finite mechanical elastic line with unfixed end (a) and of corresponding ladder filter (b)
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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 |
(9) |
at the
stop band, el
>1 |
(10) |
and at
the cutoff frequency, el =1 |
(11) |
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 |
(12) |
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