V.2 No 1

3 (appendix)

Respond to Dr Scanlan on ladder filters

Then let us analyse, what can the Reviewer do with his matrix

To keep the condition (1), the transmission constant of his chain has to be the same along the ladder filter. If opposite, (1) would take the form

and the Reviewer would not yield (2)- (6) from this condition.

Actually, the transmission constant of ladder filter is determined from the equality [1, p. 577] as

From the very construction of the chain of equalities in (20) we see that it is true both for matched and mismatched load. From this, the standard representation

is also true both for matched and mismatched load, though usually it is thought otherwise. But we can represent the A-matrix of ladder filter as (1) only in case if the transmission constants of all chains are equal - i.e., only at

From this we directly yield

and from this last - the condition (1) which the reviewer wrote.

Still scientists could not theoretically corroborate (22) true, in lack of theoretical grounds for it. With our method and its solutions corroborated experimentally we can easily check, how this condition is satisfied for mismatched ladder filters.

As it is commonly known (see, for example, [1, p. 542]),

The condition at which we find  A₁₁   corresponds to

The exact analytical solution for the pass band is described by (26) of our paper:

At the condition (25), this expression transforms to

From this

We see from (28) that the transmission constants of the chains are inconstant along the ladder filter and are changed with the index i   as trigonometric tangent. It straight follows from this that while (20) and (21) are true, (22) is untrue even for a ladder filter with identical chains. And this is justified, as the very condition (24) of unloaded chain is the condition of mismatched filter; consequently, we have to seek the transmission constant for the mismatched filter. With it, the load will affect not only the last chain. This affection will be nonlinearly distributed along the filter and will mismatch all its chains. The transmission constants will become inequal, which we see in (23).

Thus, neither Reviewer nor you could find the transformation which would enable you to pass from Reviewer's (1) to our (26)- (28), since the point is not in variations of methods, as you used to read in submitted manuscripts, but in basic distinction of approach to mismatched ladder filters calculation. We provided this in our work. In the item 3 we showed incorrect to seek the expression for the wave impedance of the chain of filter from the condition of shorting and open output in the view of matched load. You confined yourself to the statement of the fact that we consider these regimes. Though this conclusion per se has a great practical and theoretical importance for the circuit theory, as the correctness of calculations is based on it. And if this conclusion was unprofitable or inconvenient for someone, this does not change the situation and the calculations will not become more accurate, will they?