download searchable PDF of Circuit Design book - IEEE Global ...

246 Linear Amplifier Design Tools
T=0.4977 / -89.7372°, R=0.4977 /90.2628°, and Z,=49.9932+j49.53670
ohms. These results are shown in Figure 7.9. An application might be tuning
the network by varying a reactance in the series branch.
7.2.4. Summary ofImpedance Mapping. The linear, three-port network was
characterized by its scattering parameters; a set of equivalent two-port parameters
was then obtained. Besides applications in ladder analysis, the results
clearly show the bilinear effect of any network branch on network response.
This is true because any two-port network branch may be "brought out" as a
third port. Hand-held computer Program A7-2 was provided to make the
three-port to two-port reduction calculations.
Although the three-port to two-port reduction showed the bilinear effect of
branch scattering parameters on scattering responses, it was necessary to show
that bilinear functions of bilinear functions are bilinear; this was illustrated.
The bilinear theorem was thus proved. This theorem states that every Z, Y, or
S response of a linear network is a bilinear function of any branch impedance,
admittance, or scattering parameter, in any mixed association. A neutralization
example was worked. The bilinear theorem has a lot to do with feedback
analysis, especially when applied in conjunction with mapping. For example,
transistor shunt feedback is easily analyzed by this technique.
It was noted that the standard bilinear form may be decomposed in several
different ways. For network behavior, it is especially useful to mold it into a
form having a complex constant for translation added to an orientation factor
that multiplies the generalized Smith chart function. In this way, the effect of
all possible values of a branch impedance on a network response may be
visualized. Since bilinear transformations map circles and lines into circles and
lines in mixed association, certain critical branch loci can be visualized in the
response plane for subsequent analysis. Hand-held computer Program A7-3
was provided to convert two-port scattering parameters into bilinear coefficients
that relate normalized load impedance to input reflection. In addition,
the mapping coefficients were calculated. Three examples of this technique
were provided, and Smith charts illustrated the results.
7.3. Two-Port Impedance and Power Models
The development in this section will be in terms of admittance parameters.
One reason for this is that a recently defined power gain is developed in these
terms. Impedance could have been used just as readily for the general aspects;
in fact, most equations in this section can be expressed using impedance
parameters by simply replacing all the y's and Y's by z's and Z's, respectively,
and exchanging V's and I's. Ironically, there has been a great emphasis on
scattering parameter relationships, and almost all recent design aids involve
these parameters. However, many crucial concepts are more readily seen in