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372 Other Direct Filter Design Methods
and a minimum radius, defined by
M-R
gmin= I-MR' R is fixed. (9.71 )
II
The image encircles the origin when M < R. This is illustrated in Figures 9.35
and 9.36. Although it is incidental to the following application, angle ~ in
Figure 9.36 is
(9.72)
9.5.1. Power Bounds Between a Complex Source and Loads. Figure 3.3 in
Section 3.2.3 illustrated the connection of a fixed, complex source to an
arbitrary load impedance. Example 7.1 in Section 7.1.2 illustrated the calculation
of power transferred to load impedances contained within and on a 2: I
SWR circle. The solution was based on the generalized Smith' chart situation
shown in Figure 7.2. The circuit is reproduced in Figure 9.37.
Compact expressions bounding the power delivered may be obtained by
applying the bilinear mapping result in Section 9.5.1. Suppose that both the
source and load standing-wave ratios Ss and SL' respectively, are defined with
respect to resistance Ro. Then the f plane in Figure 9.35 becomes the load
Smith chart when Z=ZL' Z,= R o , and Z.=Z,. The reflection magnitude is
related to the standing-wave ratio S by
I fl=S-1
(9.73)
S+l'
Applying (9.66) to this case yields
S -I
R=lfd~_L­
(9.74)
SL +1 '
and (9.69) yields
S -I
M = If,1 = S:+ I . (9.75)
According to (9.67) and (3.47), when Igi IS maximum, load power is
minimum, and vice versa. So (9.70) yields
. P L 4S L
S,
mm-= (9.76)
Pa. (S,SL + 1)2
+
E, _
Figure 9.37.
A fixed complex source connected to arbitrary complex loads.