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242 Linear Amplifter Design Tools
Figure 7.5.
R,w
The bilinear transformation w=T+Rp,
(Y = I/Z) always map circles and lines into circles and lines. For bilinear
transformations in linear networks, there is a much more useful decomposition
of the standard bilinear form in (7.1); this is
Z-Z
w=T+R Z
+ Z
;' (7.38)
In (7.38), T locates the center of the branch-image circle, the magnitude of R
scales its size, and the angle of R determines its rotation with respect to the
w-plane coordinate system. The branch-image Smith chart has a complex
normalizing impedance (Z,), as explained in Section 7.1.2. This is illustrated in
Figure 7.5.
The significance of (7.38) and Figure 7.5 stems from the bilinear theorem in
Section 7.2.2. The small, rotated Smith chart represents the entire right-half
plane .of any linear network branch impedance, admittance, or scattering
parameter. The w plane represents the network's response function, expressed
as a scattering, impedance, or admittance parameter. So the w plane might be
the Zin plane, the Yin plane, the S\2 plane, etc. The small branch-image circle
mayor may not fall within the w-plane unit circle, should that be a scattering
parameter and therefore relevant.
It is not difficult to find expressions for the complex constants T, R, and
Z,. These relationships are obtained by putting (7.38) into the form of (7.1)
and comparing the coefficients of Z. Thus (7.38) becomes
It is seen that
and
Z[(T+ R)/Zn + [T- R(ZJZn]
w= Z(I/Zn+ 1 (7.39)
(7.40)
a 1 R=--T.
a,
(7.41)