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Load Effects on Passive Networks 371
This form is unique for any bilinear transformation that maps the right-half
plane onto a unit circle. Another such form shown in Figure 9.35 is
Z-Z*
g=G(Z)~--g.
Z+Zg
(9.67)
The purpose of this section is to introduce the fact that bilinear transformations
that map unit circles onto unit circles have certain elementary properties.
For instance, they must have the form
. f-f o
g= H(f)=eJY-­
1- ff~ ,
(9.68)
where Ifol < I (see Cuthbert, 1980). According to (9.67), the g-plane origin is
the image of Z = Z;. But (9.68) shows that it also corresponds to f = fo.
Therefore,
Z;-ZI' ~ .
f o= F(Z*) g = Z*+Z = Me J •
.
(9.69)
Cuthbert (1980) shows that, for a fixed R in (9.66) and Figure 9.35, the
constant-Ifl-circle image in the g plane has a maximum radius, defined by
g
f
R is fixed, (9.70)
9 Plane
GIF"lfllj
I
I
Reg
I
I I 1m,
Z Plane
,
I
I
J ,
I G~ • Z,
J I
I
I
I I
I
I I ReZ
\ \
\
F(Z;I
\ \ 1m f
\ \ eZj
\ ,
\ \
\ \
\ \ t
Ref
Lz forlfl=R
f Plane
Figure 9.36.
Some details of the transformation between unit circles.