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Narrow-BaruJ L, T, aruJ Pi Networks 181
plane. Normalizing impedances to resistance R" reflection coefficient (4.57) is
rewritten as the bilinear function
Z/R,-I
p= Z/R, + I . (6.22)
The mapping is illustrated in Figure 6.10. Lines of constant resistance map
into closed circles of constant, normalized resistance in the Smith chart, and
lines of constant reactance map into circular arcs of constant, normalized
reactance. On a Smith chart with its center normalized to impedance I+jO
ohms (or mhos) according to (6.22), any complex number has its inverse
appear symmetrically about the origin; i.e., given a point Z/R p
the point
YXR, appears on the opposite radial at the same radius, where Y= I/Z. An
easily read summary of Smith chart properties has been given by Fisk (1970).
The first result of Example 6.1 is plotted in Figure 6.10. The normalized
reactance (X 2 /R, = 10.68/25=0.43) amounts to a displacement of +0.43
along the normalized constant resistance circle (6/25=0.24). Then, since the
X, matching reactance is a shunt element, the impedance point is converted to
an admittance point by reflection about the origin, as shown in Figure 6.10.
This point is necessarily on the normalized unit circle passing through the
center of the chart (the center representing R, = 25 +jO ohms). Now the Smith
chart is considered an admittance chart instead of an impedance chart. Thus
the displacement due to X, = - 14.05 ohms is considered a normalized susceptance
of + 1/14.05x25= + 1.78 mhos, which carries the transformation to
the chart center, as required. The reader should plot the second solution of
+'
jX
Constant X
z
R,
R,
R
x' -.5
Constant R
-1
I I I I I I I I I I I I I I I I I I I I I
1.0 0.8 0.6 0.4 0.2 0 0.2 DA 0.6 0.8 1.0
Figure 6.10. The ordinary Smith chart (the unit reflection coefficient circle) on the left is a map
of the right-half Z plane on the right.