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Prototype Network 279
element values. This equivalence is further developed in the work of Matthaei
et al. (1964, pp. l44-149); it will also be apparent from the development in the
next section.
8.1.3. Prototype Network Selectivity. According to (4.9), the ABCD (T)
matrix of an entire ladder network can be obtained by multiplying the ABCD
matrices of the component subnetworks in order. For example, (8.6) and (8.8)
may be applied for the N = 2 (two-resonator) network in Figure 8.6. The result
lS
T=[j Z OI2 Y 2 j Z 012 ].
jYIZOI2Y2+jYOI2 jYIZOl2
(8.10)
If another inverter matrix and resonator matrix are appended to (8.10), the T
matrix for the N = 3 prototype network corresponding to Figure 8.1 may be
obtained. Only the resulting C element is of interest:
C= - YIY2Y,Zo12Zo2' - Y,YOl2Zo2,-YIZOI2Y02J'
Similarly, the C element for N =4 is
C= - YIY2Y,Y.Z012Z023Z0'. - jYIY2Z012Z02JYO'.
(8.11)
- jYIY.Z012Y02JZO'. - jY,Y.Y012Z02,ZoJ. - jY012Z023YO'.' (8;12)
There are 52 terms in the C element of the T matrix for N = 5. The need for
only the C element is explained next, followed by the identification of the
transfer function's complex polynomial in frequency variable F, defined in
(8.2).
Assume that the terminals of the overall two-port network are located as
shown in Figure 8.1; i.e., the load resistance and the source shunt resistance (if
it exists) are included inside the network as parts of the terminal resonators.
Thus output current IN=O. Then the ABCD-defining equation (4.8) shows
that II/VN= C. The loss function of interest is
L( )= VN(wo)
W VN(w)'
(8.13)
Since II(w)= II(wo), it follows that the desired loss function is equivalent to
C(w)
L(w)= C(wo) . (8.14)
Therefore, only the chain parameter C of an N-resonator filter is required for
the loss function.
A general expression for the loss function may be deduced by considering
the N =3 case in (8.11). Assume that the sOurce includes a nonzero conductance.
From (8.1), there will be two resonator admittance terms that have both
real and imaginary parts, namely Y I and Y,. Resonator admittance Y2
= jFIX,. The first term in (8.11) will produce frequency variable jF with
exponents 3 and 2. The second and third terms in (8.11) will produce