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Continued Fraction Expansion 51
Note that the magnitude symbols have been omitted in the last two equations
and that the substitution s= jw has been made on the assumption that
magnitude functions such as (3.50) are involved. Also, the grouping of
parameters is strategic, because it can be shown that, for lossless networks, A
and D are even functions of s, while Band C are odd. Further, reciprocity
requires that AD - BC= I. Beyond that, the grouping is convenient because
adding or subtracting Hand K cause major cancellations. One good reason
for defining K at all is the following important result:
(Ho+ Ko)RJR,],
(H,-K,)R,
(3.70)
where the e subscript denotes an even polynomial and 0
an odd polynomial.
Example 3.11. In Example 3.10, K(s) was given in (3.57) and the numerator
of H(s) was found as (3.60). Note that the denominators of Hand K are the
same. Enter the numerators of Hand K into Program B3-4 in that order; then
(3.70) yields the ABCD matrix numerators without difficulty. The result is:
[~ ~]=
102s' + 24
4/3(s'+4)
114s 3 +72s
4/3(s'+4)
136s 3 + 96s
4/3(s'+4)
152s 4 + 168s'+32
4/3(s'+4)
(3.71 )
where R J
= R, = I is assumed, as explained in Section 3.4.4.
3.3.3. Summary of Polynomial Addition and Subtraction of Parts. This section
began with a simple BASIC language program to add and subtract even,
odd, or all parts of polynomials. It continued with a look at the well-known
ABCD (chain) parameters for two-port networks. The H(s) and K(s) functions
were related to the ABCD parameters by considering input power transfer and
input impedance and then assuming s=jw for implied magnitude functions.
The right tools make the task quite simple along theoretical lines that are easy
to remember after a little practice.
The strategy behind the convenient ABCD development is to obtain simple
expressions for LC impedance and admittance parameters in terms of the
ABCD polynomials already found. A continued fraction expansion of these
produces the corresponding network element values, as shown next.
3.4. Continued Fraction Expansion
Continued fraction expansion of reactance functions (ZLcl will be described
and used to realize a lowpass network as the last step in the LC network
synthesis procedure. These functions are the port impedance or admittance of