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Impedance Mapping 239
1 3
al~ ~a3
[SiI J
Figure 7.3.
1 i
2
b, '2
-1',-
2
A linear, three-pori network with scattering parameters Sij.
ters will be upper-case Su (see the related network in Figure 7.3). Port 2 is
constrained by the reflection coefficient f, so that
a,=f,b,. (7.22)
Making this substitution in (7.21) yields
hI = sllal + sl2f 2hZ + sl3a3'
b 1 = sllal + S22f 2 h z + S23a 3'
b 3 = S31 a l + S23r l b l + s3Ja3 .
Solving (7.24) for b, yields
Substituting (7.26) into (7.23) and (7.25) yields
(7.23)
(7.24)
(7.25)
(7.26)
8 12 8 21 ) ( 8 12 8 23 )
b,=a, ( s,,+ l/f,-s22 +a3 S'3+ l/f,-s22 ' (7.27)
8 21 8 32 ) ( 8 23 8 32 )
b,=a, ( S3'+ 1/f'-S22 +a, ~,,+ 1/f,-S22 . (7.28)
The equivalent two-port parameters are immediately available by comparison
with the original set of two-port equations:
8 12 5 21 8 12 8 23
5"=S"+I/f_ ' S'3=S13+I/f_
2
8 22 ,2
8 22
S32S21 S32S23 (7.29)
S31 =s31 + I/f _ S33=S,,+ I/f _
2 822 2 522
In this case the two ports of interest are ports I and 3. Also note that as f,--->O,
Sij--)Sij' as required.
Program A7-2 in Appendix A for the HP-67/97 evaluates the equations in
(7.29). The nine three-port scattering parameters are input first. The program
then converts any given port-2 terminating impedance (Z,) to the reflection