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Doubly Termi/lllted Elliptic Filters 353
Amstutz's subtle step depends on the equivalence of any two networks
having the same topology but permuted trap positions. The key to his method
is to permute each of the traps into the input position and apply a recursive
algorithm that yields all of the element values. An additional feature of his
type-s symmetric filter program is the calculation of the critical unloaded Q
(uniform in both inductors and capacitors) that just erases the trap notch
nearest the passBand.
This section begins with the application of the input impedance and its
derivative with respect to frequency. The basis of the trap permutation scheme
is then described. Finally, examples using the symmetric and the antimetric
programs are discussed.
9.3.1. Input Impedanee Relationships. The analysis will employ the midseries
topology in Figure 9.120 without loss of generality. It was remarked in
Section 8.1.1 that the impedance of a series LC branch resonant at w n is
simply jwnLF (for example, see the admittance case in (8.28), Section 8.3.2).
Suppose that the (M" K,) branch in Figure 9.120 is resonant at s, =jw,. Then
the input impedance in the neighborhood of that frequency is
Z;n(S)=SL,+M,(S- :i). (9.32)
Note that impedance Z, seen to the right of the (M" K,) branch in Figure
9.120, is not zero in practical cases. Differentiating (9.32) with respect to
complex frequency s yields
Zin = L, + M,( I