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I
~
192 Impedance Matching
load impedance looking toward the load. This circumstance exactly fits the
discussion of power transfer from a complex source to a complex load in
Section 3.2.3. In Figure 6.18, the reflection coefficient Po is defined with
respect to resistance ~, and P is defined with respect to gn+ I' For lossless
networks, the power available at the source is also available at the load. From
Section 3.2.3,
Ipl=IPol, (6.43)
where
(gn+ 1 in ohms). (6.44)
Clearly, a good impedance match occurs when Zin is nearly equal to the
generator resistance gn+ I; this is precisely stated as the minimum Ipl· Over a
frequency band, a good impedance match would be obtained by minimizing
the maximum Ipl. This is shown in Figure 6.19.
Fano (1950) stated the theoretical limitation for load networks representable
as resistively terminated LC networks, such as in Figure 6.16. Their
lowpass form is
for single-reactance loads, and
(00w'ln..Ldw= :!!-(~_1)
)0. Ipi gj g, 3
(6.45)
(6.46)
for two-reactance loads. Note that the integrand is essentially the return loss
in (4.58). The interpretation for the gain-bandwidth limitation describe,! by
(6.45) is illustrated in Figure 6.19 for the bandpass case: the reflection
magnitude may be low (good match) over a narrow band or higher (poor
match) over a wider band. Fano noted that in no event should the reflection
magnitude in the band be zero, as is commonly'the case with filters. Making
I/lpl very large at any point in the pass band necessarily reduces the
bandwidth because of the inefficient use of ihe areas in the integrals above.
Case t:
Narrow
band
I
I
c 0
'0
'" ..0 ~
o'
u
c " i!
.. E
0
"fl 'c
~
'"
0
0 W a Wo w b
Radian frequency, W
Case 2:
Wide
band
Figure 6.19. Constant gain-bandwidth tradeoffs for a good match over a narrow bandwidth
(case I) and a poor match over a wide bandwidth (case 2). {From Matthaei el aI., 1964.]