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Bilinear Transfonnations 235
accomplished easily on desktop computers; the interested reader is referred to
Kajfez (1975).
7.1.2. Generalized Smith Chart. Using methods described by Churchill
(1960, pp. 76-77), it is a straightforward matter to show that every bilinear
transformation of the closed right-half Z plane onto a closed unit circle must
have the form
(7.15)
where f3 is real and Re(Z,)>O. The last requirement is especially emphasized.
The exponential term merely rotates the unit-circle image and will henceforth
be dropped. The generalized Smith chart maps impedances according to
Z-Z, Z-jX,-R,
p=--.=., (7.16)
Z+Z, Z-JX,+R,
where Z=R+jX and Z,=R,+jX,. Clearly, (7.16) could be normalized to R,
by division in the numerator and denominator in the fashion of the ordinary
Smith chart relationship given previously in (6.22). The obvious remaining
difference is the term jX,. A little thought shows that it may be combined into
a new reactance component (X - Xc> instead of the usual X component. The
generalized Smith chart then represents the ordinary chart with center Z, and
constant reactance lines (X - X,). One practical application concerns power
transfer from a complex source to a complex load, as discussed in Section
3.2.3. Thus (7.16) is exactly comparable to (3.46). Note that whether Z,
appears in the numerator or in the denominator is a matter of arbitrary
definition. It is convenient here to represent the chart center as Ze-
It is also important to define the generalized reflection coefficient in
admittance form, as follows:
Y,-Y
p= Y'+ Y . (7.17)
,
The generalized Smith chart no longer allows substitution of Z= I/Y in order
to change from an impedance to an admittance basis. This does not change
(7.16) into (7.17) unless Z,= I/Y, is real.
Example 7.1. Consider a complex source connected directly to a complex
load, as in Figure 3.3 in Section 3.2.3. Suppose that Z,= 25 - j50 ohms and ZL
is defined as causing a 2 : I standing-wave ratio (SWR) on a 50-ohm transmission
line. What is the range of power delivered to the load relative to the
power available from the source? The solution will be obtained graphically
here and analytically in Section 9.5. The procedure will be to select three or
more impedance points on a 2: I SWR circle from a Smith chart normalized
to 50 ohms. Then these points will be plotted on a generalized Smith chart
normalized to the conjugate of the source impedance in accordance with the