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82 Ladder Network Analysis
many occurrences of these components in filter models, especially in microwave
filters; one of the latter will be designed and analyzed in Section 9.1.
Because of prior formulation, this topic reduces to a consideration of the
. ABCD-parameter calculations, some useful approximations, functional-form
shortcuts in the programming associated with the calculations, and additions
to the topological list technique.
4.3.1. Transmission Line ABCD Parameters. The ABCD parameters for
dissipative, uniform transmission lines have been given in many places (for
instance, Matthaei et aI., 1964, p. 28):
A=D=cosh(NP+jO),
(4.13)
B=Zosinh(NP+jO ),
C= Yosinh(NP+jO).
(4.14)
(4.15)
Real characteristic impedance Zo is the reciprocal of admittance Yo, angle 0 is
the line electrical length at some frequency w, and NP is the frequencyindependent
loss, in nepers, for that length of transmission line. Note that I
neper= 8.686 dB. The hyperbolic functions above have complex arguments.
They may be evaluated by the following indentities from Dwight (1961, pp.
153,4):
eNP(cos 0 + j sin 0) - e- NP(cos 0- j sin 0)
sinh(NP+jO)
(4.16)
2
e NP (cosO + j sin 0) + e-NP(cos 9 - j sin 0)
cosh(NP+jO)
(4.17)
2
Note the functional similarity; only one interior sign is different, so that one
program segment with a flag variable should suffice for evaluation. Programmers
of hand-held calculators should also note the efficiency of the polar-torectangular
conversion of unity at angle 9 to obtain cos(O)+jsin(O) in one
operation.
4.3.2. Lossy Transmission Line Stubs. A compact means for calculating the
input impedance of a short- or open-circuited dissipative transmission line will
be described. The analysis of Section 3.3.2, leading to (3.66), is directly
applicable for an arbitrary load impedance at port 2 of a two-port network:
AZ L
+B
Zt=CZL+D' (4.18)
where Zj is the stub input impedance, and the ABCD parameters are given in
(4.13)-(4.15). For ZL approaching infirtity and zero, it follows that
Zo
Zt.oc= tanh(NP+jO)' (4.19)
Zj.sc=Zotanh(NP+j9), (4.20)