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Input Impedaoce Syuthesis From Its Real Part 61
Table 3.5.
Open- or Short-Circuit ZLC
Impedance Functions
Pole at Origin
ZIl=PJQo
"'2~Q./Qo
Y22=P e /P o
No Pole at Origin
zll=PoIQe
"'2~Qo/Q.
Yn=Po/Pe
Example 3.16.
Consider the pi network in Figure 3.8. Suppose that the
impedance ZRLC looking back into the terminated network at port 2 is the
same as (3.100). Open-circuit impedance function Z,2 is selected because Y22
implies a short circuit that would prevent determination of C,. But the
presence of C 3
means that a port admittance function is required, so that the
selection from Table 3.5 is
zn'= Q. = 1.298316s 2 +0.482804 . (3.108)
Qo • S3 + 1.360294s
Continued fraction expansion Program B3-5 applied to (3.108) yields C, =
0.350 farad, L 2 =2.890 henrys, and C 3 =0.931 farad after scaling from the
I-ohm source to the 2.2-ohm source shown in Figure 3.8.
2.2 n 1 L2 2
C:C E
Figure 3.8.
A three-pole normalized lowpass network.
3.5.4. Impedance Real·Part SJ'IIthesis Summary. It has been shown that
Iossless networks terminated on only one end can be synthesized according to
input impedance behavior. This is based on the fact that, for constant input
current, the input power (and consequent output power) is proportional to
input resistance. A similar statement can be made concerning input conduc·
tance in the case of constant voltage sources. Singly terminated instances of
resistive sources connected to unterminated lossless networks are equivalent
by proper consideration of the reciprocity theorem.
The Gewertz procedure was described for problems beginning with the
even resistance function of frequency. Substitution of w 2 = - S2 produces a
polynomial whose denominator left-half-plane roots produce the input impedance
denominator. The input impedance numerator coefficients are obtained
by solving a system of real, linear equations involving these roots and known
coefficients. The result is the input impedance ZRLC of a terminated network.