V.2 No 1

41

Mismatched ladder filters

The same as in case of unloaded filter, at the transparency band we see n resonance peaks (taking into account the nth shorted node). The conditions of their appearance are determined now by the vanishing denominator of (16). The filter input impedance written as _sc for the filter with the shorted output also essentially differs from the results obtained by the two-port method. Its value will be determined now by the following system of expressions:

at the pass band of the filter, _el <1

(19)

at the stop band, _el >1

(20)

and at the cutoff frequency, _el =1

(21)

We see from (19) – (21) that at the pass band the input impedance will have (n –1) resonance peaks. At the stop band the impedance falls with frequency proportionally to 1/_el+, and with the boundary frequency it is equal in amplitude to the impedance only at large n.

As the results have been refined, the relation between the ladder filter natural impedance and the filter input impedance (being important for calculations) will essentially change. We can determine it by an analogy with electrical transmission lines as follows (see, e.g., [4, p. 572]):

.     

(22)

Substituting sequentially (13) – (15) and (19) – (21) into (22), we can see that in general case the conventional equality (22) is not true. It speaks that for ladder filters this method to find the natural impedance is unacceptable. Although for the single filters the conventional computations retain true, the same as the two-port method itself.

The reason of discrepancy of the results obtained with the help of DEMA and by the two-port method is the same. The same as (16) – (18), solutions (9) – (11) describe the standing wave – just the process which has to take place when the load mismatched and in this connection the waves reflected from the ends of the filter are produced. The two-port method does not take it into account. And in the finite filters we also cannot obtain the solution on the basis of a simple superposition of direct and reverse waves, since in these filters the multiple reflections from both ends take place. So, in general case, we cannot use for sure the idle run and shorting regimes to find the filter natural impedance. We have to determine it on the basis of complete analytical solutions – just so as we did it with the help of DEMA.

4. An arbitrarily loaded finite ladder filter

In ladder filters practical calculations the most important is the case when the filter is loaded arbitrarily. As we saw in the introduction, generally the two-port method cannot solve this problem, because its application requires, the filter output impedance to be matched with the load. The assemblage of the method DEMA with the method to obtain exact analytical solutions for mechanical analogues enables us to find the solutions so necessary for the practice.