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Billlteral Scattering Stability and Gain 257
conditional, i.e., when K < J. An example was worked using Program A7-4.in
Appendix A. It calculated the stability factor, maximally efficient gain, both
terminations, and the overall stability factor, which includes the damping
conductances of these terminations. A transistor model for the input and
output branches was derived from the conjugate termination immittances.
This provides a starting point for matching network problems.
Developments in this section were written in terms of admittances, although
the same development in impedance terms essentially requires only a
change in labels. Many of the concepts introduced are more easily visualized
in these parameters than in scattering parameters. Ironically, most recent
computer design aids are based on scattering parameters because this is the
most effective way to characterize physical systems accurately. The next
section develops very flexible gain relationships in terms of scattering parameters.
All of these concepts are valid in any standard set of characterizing
parameters, and conversion from one set to another is simply a matter of
running existing short computer programs.
7.4. Bilateral Scattering Stability and Gain
The scattering parameters introduced in Section 4.5.2 are easier to measure
than other network-characterizing sets, e.g.~ open-circuit z and short-circuit y
parameters. It is customary to normalize measured scattering parameters to
50+jO ohms. The measuring process then requires port termination by 50­
ohm resistors, which can be obtained with considerable accuracy at even very
high frequency. Conversely, it is extremely difficult to obtain an open or short
circuit at high frequencies. Therefore, the network will now be viewed as
illustrated in Figure 7.17.
It will be essential to renormalize the scattering parameter set from one
normalizing impedance at each port to other values. This will usually amount
to changing a port's normalizing impedance Z; from Zi = 50 + jO ohms to some
new value, Z;. The original set will be designated S, and the new set S'. The
reflection coefficients of the new terminating impedances with respect to the
original ones will be r, and r" as shown in Figure 7.17. This renormalization
will provide comprehensive expressions for network behavior between complex
sources and complex loads.
Network with S
normalized to Z; at
port 1 and normalized
to Zi at port 2
2
5;1
Figure 7.17.
A two-port network with scattering notation.
2'