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58 Some Tools and Examples 0/Filter Synthesis
I, I,
Lossless
netlNork
+
v, R,
P, P,
Figure 3.7.
A singly terminated, lossless network.
networks. A ladder network is a minimum-phase network, but a bridge circuit
is not. If the real part of the input impedance of a minimum-phase network is
known for all frequencies, then its imaginary part (reactance) is dependent
and can be found. A desktop computer program for finding the reactance at
any frequency, given the piecewise linear resistance function versus frequency,
will be furnished in Section 6.7. Here, a regular resistance function of
frequency will be given in polynomial form, and the entire Z(s) rational
function will be found. This will be the ZRLC shown in Figure 3.7. Then, a
method will be described for finding the corresponding reactance function
ZLC' so that the continued fraction realization previously given may be
employed to find the network element values.
3.5.1. Synthesis Problem Statement.
given as the rational polynomial
A+Aw 2 + ..·+Aw 2m
R()no I m
W = B o
+ B}(,J2+ ... + lw2n
Suppose that a resistance function IS
(3.94)
Such a function may result from the fitting procedure of Section 2.5. However,
note that the denominator in (3.94) has a nonunity coefficient (B o ), and the
coefficient of highest degree is unity. As mentioned earlier, resistance functions
are even, so that all powers of ware even. The goal is to find the
corresponding impedance function:
3.0 + als+ a 2 s 2 + ... + ams m
Z(s) = , (3.95)
b o +b,s+b 2 s + ... +Is·
Remarks similar to those regarding the denominator coefficients in (3.94)
apply to the denominator of (3.95). There are at least two ways to solve this
problem: Bode's method and Gewertz's method, as described by Guillemin
(1957). The latter, which follows, is more compact.
3.5.2. Gewertz Procedure to Find RLC Input Impedance. Form an even
function of complex frequency by substituting w' = - s' in the given resistance
function (3.94):
A,,-A l s'+A,S4_ ... +(-ltAms'm
Z,(s) = B '4 • ,. (3.96)
o-Bls +B,s -·.. +(-1) s