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260 Linear Amplifier Design Tools
such as a single-chip transistor, is
K>I, I~I< I, (7.97)
where stability factor K is defined by Equation (E.2), and ~ by Equation (E.l)
in Appendix E. These criteria are not sufficient for cascaded amplifiers or
active devices embedded in reactive networks, because there may be local
instability.
When the network is not unconditionally stable, the regions in the termination
planes can be located by imposing the unit reflection magnitude constraint
at the opposite port. Consider the fixed source r l and locate the values
of load r, where IS111 = I. Note that rl is a generalized reflection coefficient, as
described in Section 7.1.2. An important requirement is that the real part of
the normalizing constant be strictly positive. There are no other requirements,
so that it is possible to assume that r l is normalized to an arbitrary, positivereal
generator impedance. Then rI= 0, and it will still be true that IS'III = 1
locates the r, stability boundary. This is a simplifying argument which
supports the conclusion that the stability region in the r, plane is independent
of rl'
lt has been mentioned that rl =0 in (7.92) yields (7.32), where r,=r, in the
present analysis. Solving this for r, yields
S;\-:-Su
r, = S' S ~ .
II 22
(7.98)
Now S~ I is within a unit circle if
, Z-l
SII=Z+1 (7.99)
for a hypothetical Z, which is introduced so that impedance mapping may be
employed. This is illustrated in Figure 7.18. Substituting (7.99) into (7.98),
alZ+a,
r,= a 3
Z+ I '
(7.100)
.--"-+"7"':::---.:511 image
a 1 Z + a 2
'2"" a 3 Z+1
Hypothetical
Z plane
511
plane
• z;
Figure 7.18.
The r2, S'll, and hypothetical Z planes for deriving the stability region.