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Continued Fraction Expansion
It was shown in Section 3.3.2 how to find the rational polynomials for A, B,
C, and D. The ABCD linear equations were given in (3.62) and (3.63),
comparable to the z- and y-parameter equations above. In order to find a port
immittance (impedance or admittance) as functions of ABCD when the
opposite port is terminated by either a short or open circuit, it is necessary to
find the z and y parameters in terms of the ABCD parameters. For example,
solve for V2 in (3.63):
Using this in (3.62) yields
VI=II(~)+I2(D~ -B).
ss
(3.83)
(3.84)
Comparison of (3.84) with (3.77) provides z" and z12 in terms of the ABCD
parameters. The coefficient of I, in (3.84) is further simplified for lossless two
ports because AD- BC= I in that case. Therefore, the following identities
apply for two-port networks:
Z = i: [~ ~l (3.85)
Y =i [_~ -U (3.86)
These are valid for complex numbers or for rational functions; the latter will
illustrate the last step in network synthesis. For example, (3.85) says that the
open-circuit impedance parameter z" = AIC and both A(s) and C(s) were
described in terms of H(s) and K(s) in (3.70). The numerator and denominator
polynomials of Hand K were defined in (3.54) and (3.55). For ZII' the result is
e,(s)+f,(s)
z,,(s) = RI eo(s) _ fo(s) . (3.87)
This is the impedance for Case I, N =4, in Figure 3.5. Note that I/Y" was
relevant to N=3 but not to N=4. An example from Ternes and Mitra (1973)
is given below.
Example 3.14. Given the characteristic function K=S3, find z" and I/y"
and the related networks. It is seen from (3.55) that f(s) = S3 and p(s) = 1. Then
(3.56) is
e(s)e( - s)= 1-S6= (I + s)(I- s)(1 + s+ s')(1- s+ s2). (3.88)
As noted in Section 3.2.4, the roots of e(s) are the natural modes, which must
be in the left-balf plane. Therefore, the transducer function according to (3.54)
IS
e(s)
H(s)= - = (I + s)(1 + S+S2) = I + 2s+ 2s 2 +S 3 .
p(s)
(3.89)