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52 Some Tools and Examples 0/ Filter Synthesis
lossless LC networks when open or short-circuited at the opposite port (see
Figure 3.4). They are rational polynomials that are always an even polynomial
over an odd polynomial or vice versa. Such expansions provide lowpass or
highpass network element values and can also be used 'to determine if a given
polynomial is a Hurwitz polynomial (all roots in the left-half plane).
3.4.1. Lowpass and Highpass Expansions. Continued fraction expansions
may be finite or infinite. Two finite examples and their equivalent rational
polynomials are:
Z,(s)=2s+
I
3s+ I
4s+­ Ss
Z2(s) = ...L + --:--~--:--
2s ...L + I
3s ...L+_I_
4s I/Ss
120s· + 36s 2 + I
60s 3 + 8s
(3.72)
1+ 38s 2 + 120s·
2s+ 64s 3 (3.73)
A convenient shorthand for representing continued fraction expansions IS
described by Vlach (1969); applied to (3.72) it is
1 I I
Z,(s)=2s+ -3 -4 -S' (3.74)
s + s + s
3.4.2. A Continued Fraction Expansion Program. Consider the rational polynomial
to be in one of the following forms or their reciprocals:
aO+a2s2+a4s4+ ... +ans n (3.76)
n IS even;
(3.7S)
ao+a 2 s 2 +a 4 s 4 + ... +an_1s n - 1
als+a3s3+ass5+ .... +ans n
n is odd.
Program B3-S in Appendix B is adapted from Vlach (1969); it requires only
lines 210-340 for computation.
Example 3.13. Program B3-S will be run using (3.72) for the cases where the
rational polynomial represents an LC. two-port input impedance with an
open-circuit load or an input admittance with a short-circuit load. Consider
Case I for maximum degree N = 4 in Figure 3.S. Certainly, the input impedance
of the network shown must be Z=sL+remainder, according to the form
of (3.72). Therefore, the first element must be an inductor with a value of 2
henrys. If the remainder polynomial Z, is inverted to provide Y,= I/Z" then
the next term removed must be Y=sC, where C=3 farads. Comparison of
this case with (3.72) shows how each element value was obtained for the
lowpass network. Note that a short circuit across the S-farad capacitor would