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Polynomials From Complex Zeros and Products 47
R,
"
Lossless
network
Z, P2
Figure 3.4.
A doubly terminated lossless network.
+
v,
(3.48)
(3.49)
is the input impedance Z, for the lossless network in Figure 3.4. The network's
load termination is R,; the resistive terminations at both ends make this a
doubly terminated network. Any power that enters the network must exit into
R" so that the reflection coefficient of interest is
Z,-R,
p= .
Z,+R,
Consequently, power delivered to Z, and R, in Figure 3.4 is described by the
famous Feldtkeller energy equation:
where H is the transducer function.
' )1'_ P,,- 1
IH( JW -----­
P, l-lp!2'
3.2.4. Network Synthesis Polynomials. The network in Figure 3.4 is now
assumed to be composed of lumped inductors and capacitors, so that Z, and
other impedances will be well-behaved functions of complex Laplace frequency
s. If Z(w)=R(w)+jX(w), then one should know that R(w) is always an
even function of wand that X(w) is always an odd function of w. Thus, brief
consideration will lead to the conclusion that Z*Gw)=Z( -jw). The imaginary
axis in the s variable is jw. According to the analytic continuation principle, jw
may be replaced by s in expressions where it occurs. Furthermore, the
resulting functions of s have significance over the entire s plane. This concept
leads to an identity with considerably greater importance than is at first
apparent:
IfOw)I' = f(s)f( - s), s=Jw. (3.50)
This is the squared-magnitude function, and it is also an even function of w.
Example 3.9.
Suppose that a given function is
f(s) = -76s 4 + 1Is 3 -33s'+ 12s-4. (3.51)
Compute f(s)f( - s) using Program B3-3, and save the result. Note that the
resulting magnitude function is even in s.