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Z40
Linear Amplifier Design Tools
coefficient f, with respect to 50 ohms and finds the equivalent two-port
parameters.
Example 7.1.
Consider the three-port scattering matrix
0.862 1-63~ 0.800 1160° 0.236 /75.3° ]
S, = 0.050 /20° 0.500 I-60° 0.300 I-98°
[
2.344 1129° 0.400 1100° 0.708 I - 16.1 °
(7.30)
Load these polar data into Program A7-2 by pressing keys fa and responding
to the row/column subscripts displayed. Write these data on a magnetic card
for later use. Now terminate port 2 (Figure 7.3) with a resistor of 200 10°
ohms by pressing 0, "enter," 200, and key A. The results are
SII = 0.8435 1- 64.32 ° ,
S" = 2.3561 1128.8°,
S13=0.3847 163.03~,
S33 = 0.7890 1- 15.99° .
(7.31 )
Since port 2 is normalized to 50 ohms, inputting this value will show that
5 ij = sij' as expected.
The ladder analysis method from Section 4.2 can incorporate cascaded
three-port networks having the third port terminated. It is only necessary to
evaluate equations (7.29) and convert the scattering parameters into ABCD
parameters.
7.1.2. The Bilinear Theorem. According to Penfield et al. (1970), the bilinear
theorem states that any Z, Y, or S response of a linear network is a bilinear
function of any network branch impedance, admittance, or scattering parameter.
The response and branch types can be mixed. This is evident from the
preceding three-port to two-port conversion results, as will now be shown.
Problem 4.11 asked for the input reflection coefficient in terms of the
two-port scattering parameters and an arbitrary load (port-2) reflection coefficient.
Figure 4.16 might represent such a network. As in the analysis leading
to (3.101) for input impedance, the input reflection parameter is
• _ S\2S2I f,(-il)+(+Sll)
Sll-Sll+ I/f,-S" f,(-S22)+1 (7.32)
where the two-port scattering parameter matrix determinant is
il=SIIS22-St2S'I' (7.33)
The bilinear theorem can be confirmed by comparison of (7.32) and
equations (7.29). Each of the latter have the same bilinear form as seen in the
former, at least for reflection parameters. But it is well known that a bilinear
function of a bilinear function is itself bilinear. For example, suppose that
output port-2 reflection coefficient is
z,-I
f,=-- (7.34)
z, + I '