download searchable PDF of Circuit Design book - IEEE Global ...

Bilateral Scattering Stability and Gain 261
where the bilinear coefficients are
S,,-1 SII+I A-S 22
a l = A+S 22
" a2= A+S", a,= A+S'2 . (7.101)
Now r2 corresponds to w in the standard mapping form, (7.38) in Section
7.2.3. From (7.42),
(SII + 1)(A* - Si2) + (SJJ - I)(A* + Si2)
T= (A*+S:l',)(A-S 22
)+(A+S 22
)(A*-Si2)' (7.102)
Further complex algebra reduces this to
S!2- SllL.\.*
T= -:":IS=22""'12--7: IA
-';-12
(7.103)
Since only the magnitude of mapping coefficient R will be of interest, (7.41)
will be rewritten in the form
Substitution of (7.101) yields
(7.104)
a 3 IA+S"I' (-SI2S21)
- R =-'------"'-;, "':'-c;;"-'--"::
a! (A+s 22 )2Is 2,1'-IAI 2 (7.105)
The stability circles have been located in the r2 plane, as illustrated in
Figure 7.18. Their center is located by mapping coefficient T in (7.103); this is
the complex constant expressed by Equation (E.16). The radius of the stability
circle is the magnitude of (7.105); this is (E.18). An entirely similar analysis
that locates the stability circle in the input (r l ) plane is based on assuming that
r2=0 and setting /S;21 equal to I, using (7.95). The same result is obtained,
except for interchanging subscripts. The location and radius are given by
(E.15) and (E.l7).
Bodway (1967) discussed the six possible locations for the stability circle in
a port's reflection plane, as shown in Figure 7.19. In each case, the small
Smith chart represents the port reflection plane rio For discussion, suppose
that the stability circles are in the output plane and the Smith charts represent
the r2 output termination plane. Then their interior represents passive terminations
having RL>O. In the left-hand cases (a-c), the port origin is not
enclosed by the stability circle; therefore, the device is stable outside the
stability circle, corresponding to positive input resistance. In the right-hand
cases (d-f), the origin is enclosed by the stability circle; therefore, the device
is stable inside the stability circle, especially for the positive R;n that results
from ZL = 50 + jO ohms.
In the next section, device power efficiency (or gain) '1 will be defined in
(7.108) as the ratio of power delivered to the load divided by the input power
delivered by the source. Amplifiers are usually designed so that both load