SELF

40

S.B. Karavashkin and O.N. Karavashkina

The input impedance of the filter for which, in case of the load absence at its end, we will write _oc is equal:

at the pass band, _el <1

at the stop band, _el >1   

and at the cutoff frequency, _el =1

At the pass band, dependently on ₁ and ₂ , the impedance R_oc can be active, inductive or capacitive. But the main, in any case its amplitude-frequency characteristic will have also n resonances. At the stop band with the frequency growth the input impedance monotonously tends to zero proportionally to ~ 1/_el+. And only at the cutoff frequency and at large n the input impedance is approximately equal in its amplitude to the impedance (- ₁₂)^1/2. It essentially differs from the results obtained by the two-port method (see, e.g., [4, p. 606]. One more important feature of the presented solutions is that the vibration amplitude of the last filter section (i = n) is not maximal, as we used to think by the analogy with electrical distributed transmission lines. According to (9), it differs by the multiplier  cos _el  and diminishes to zero with the vibration frequency approaching to that critical  ( _el / 2).

Fig. 3. The schematic diagram of a finite mechanical elastic line with fixed end (a) and of corresponding ladder filter with shorted output (b)

For a ladder filter with the shorted output the situation will be similar. In Fig. 3a we show the model of a finite mechanical line whose nth element is fixed, and in Fig. 3b – the corresponding diagram of an electric ladder filter with the shortened output. Noting [11] and relations (8), we can describe the process in the studied filter by the following system of expressions:

at the pass band of the filter, _el <1

at the stop band, _el >1

at the pass band of the filter, _el =1