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Bilateral Scattering Stability and Gain 263
power and input power are positive, corresponding to positive R L
and positive
R.n, respectively. Regions having positive or negative 'I are shown in Figure
7.19 by superscripts. Note that the region outside the Smith chart in Figure
7.19b represents power going into both input and output ports, since R L O. Figure 7.19c represents a backward amplifier within the small
stability circle, because both R.n and R L are negative. The shaded areas
represent loci where 'I is negative and R;n < 0 when R L
>0; this does not occur
in Figure 7.19c, f, where the device is unconditionally stable.
The Smith chart origins usually represent 50+jO ohms; the system must be
stable for 50-ohm terminations in order to be measurable. When the stability
circle does not enclose the origin of the termination's Smith chart (r;), it
defines a region of negative-real network terminal impedance (Z;n or Zq), as
shown in Figure 7.19a-c. For example, Figure 7.19c shows that only certain
negative-real load impedances could produce negative-real input impedances.
Conversely, when the stability circle does enclose the origin of the termination's
Smith chart (r;), it defines a region of positive-real network terminal
impedance (Z;n or Zq), as shown in Figure 7.19d-f. For example, Figure 7.19f
shows that any load with reflection magnitude just slightly greater than unity
will Cause negative-real input impedance. It is interesting to note that conjugate-image
matching is always possible for unconditionally stable networks,
but this mayor may not be possible for conditionally stable networks.
Example 7.9. Suppose that the device scattering parameters have been measured
on a 50-ohm system and found to be
S= [0.385 /-55°
2.7 /78°
0.045 /90° ]
0.89 / - 26.5° .
(7.106)
Using (E.2), stability factor K=0.909, so that the stability circles are of
interest. Using (E.15)-(E.18), the output plane stability circle is centered at
r,,= 1.178 /29.88°, with radius p'2=0.193; the input plane stability circle is
centered at r,1 = 8.372 / - 57.6°, with radius P,I = 9.271. These circles are
plotted in Figure 7.21, in the following section.
7.4.3. Bilateral Gains and TennilllJtions. There are several useful gain expressions
applicable to Figure 7.17 that are available from the renormalized
scattering parameters in (7.92)-(7.95). Transducer gain G T has been defined in
(7.96). Some simplification of the magnitude of (7.94) is available:
IS'II'(1-lr,1 2 )(1 -lr,1 2 )
2 •
II-rIS" - r,S22+rlr,~1
(7.107)
Using (4.51), it is easy to show that the efficiency may be expressed as
'I
IS,,,'
I-IS;,I' .
(7.108)