V.2 No 1

43

Mismatched ladder filters

In the case _load = corresponding to M = 0, (26) – (28) transform into (9) – (11). At _load = 0 (M = ) the solutions (26) – (28) transform into those similar (16) –(18) at the condition that the output of the filter (n + 1)th node is shorted. At _load = ₂ the solutions (26) – (28) also transform to those like (9) – (11) at the conditions of an unloaded (n + 1)th node of the ladder filter. And only at the condition

or

(29)

unrealisable in frames of the existing circuitry, the conditions for a pure progressive wave along-the-circuit propagation take place at all the pass band, and (26) – (28) take the form generalising the solutions for semi-finite ladder filters that was presented in [6]. In particular, for the negative delay phase these solutions take the following form:

(30)

In all other cases, a complex superposition of the progressive and standing waves will take place in the filter. Or rather, the conventional formulation of such type of superposition does not reflect enough completely the essence of processes, since the expression (26) more corresponds to the complex superposition of two standing waves. However, further we will show on a specific example that the signal phase at the filter calculated nodes (i = 1, 2, … , n + 1) generally vanishes only in a countable number of points of the amplitude-frequency characteristic, evidencing the standing wave arising in the filter. Out of these points, dependently on the frequency and number of the studied node of the filter, the phase may be as delaying as advancing with the retaining resonance peaks of the signal amplitude.

Another feature of the presented solutions reveals at the complex value of _el , i.e., in case when the ladder filter cannot be presented as an ideal filter of the low or high frequencies. With it the relation between _el and the unity losses its sense, and in the filter only the regime described by (26) can be realised, while (26) becomes true for all the range from zero to infinity. None the less, as the experience of mechanical elastic lines calculation shows and as we will show further for electrical filters, the regularity (26) at complex _el completely describes the processes both at the pass and stop bands of the filter. Only at Im _el = 0 the expression (26) losses its sense out of the pass band, and the solution takes the form (26) – (28).

To demonstrate more visually the described features of the solutions, determine the input impedance of a ladder r_LLC filter shown in Fig. 5. We choose this circuit because of a few reasons. It is quite simple, lest to cram the investigation with the additional factors. On the other hand, such circuits are often used for signal delay artificial lines. And by the conventional concept, the matched load of the low-frequency filters must be active, which makes the analysis more associative. At the same time, the changes introduced to the circuit comparing with ideal LC filters are sufficient to reveal the described effects, and the circuit itself cannot be calculated directly by the two-port method, because it has the mismatched input and output.