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and the imaginary parts yield
b ll + B Ms
gil
Therefore, the terminations must be
and
Two-Port Impedance and Power Models
YM,= g"O,+j( - b" + g"O,),
253
(7.71)
(7.72)
YML=g220,+j(-b22+g220J (7.73)
To find the 0, and 0, functions, substitute the last two equations into (7.69):
Clearly, the imaginary part of (7.74) is
The real part of (7.74) yields
Y2,y'2 = g"g2'[ (I - 0; - 0;) +j20,l (7.74)
0= Im(Y2,y'2)
, 2g 11
g 22
'
n2= I ~ Re(Y2,y'2) _ Im 2 (Y2'Y'2)
U r 2 .
g"g" (2g"g,,)
(7.75)
(7.76)
The following expression for 0, can be shown to be equivalent to (7.76) by
substituting the definition of K from (7.57). A concise expression for 0, is
The conjugate-image admittances thus are
n __ IY21ydjK 2 - 1
v (7.77)
, 2g 11
g" .
. [ Im(Y"Y12)] .
YM,=gllO,+J-bll + 2 =gllO,+JBM"
g'2
. [ 1m (Y2'Y 12) J
YML =g220,+J-b22 + 2
'
gil
(7.78)
(7.79)
where 0, and B M
, are defined by (7.77) and (7.58), respectively. Now it is seen
that (7.65) and (7.78) are conjugates. The conclusion is that the load admittance
that enables a conjugate-image match is also the load admittance that
causes the maximum possible efficiency. It is again stated that the maximum
possible efficiency is independent of the actual source admittance. However, it
is common practice to assume the source admittance YMs so that the maxi·
mum possible efficiency is also the maximum possible transducer gain. Also, it
is repeated that '1",,, has no meaning unless IKI> I.
Finally, it is noted that loci of constant efficiency in the "'" plane are an
eccentric family of circles. These may be visualized as the intersections of
inclined planes and the paraboloid in Figures 7.11 and 7.14 projected onto the