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Two-Port Impedance and Power Models 247
the impedance or admittance planes; the conversion of expressions using these
parameters to those using scattering parameters in no way changes the
phenomena. Therefore.. this section will utilize admittance parameters, and
Figure 7.1 will apply. Section 7.4 will utilize scattering parameters.
The mapping concept plays a critical role in avoiding a tangle of complex
algebra that can only obscure significant results. Its embodiment of the
generalized Smith chart is important. because the normalized power delivered
by a complex source or at the output terminals of any linear two-port network
happens to be a parabola of revolution (paraboloid) having the Smith chart as
its base. When that Smith chart is mapped into the input plane. the inclined
plane that represents input power intersects with the paraboloid of output
power. Then efficiency (output power divided by input power) is easy to
visualize, as is the point of maximum efficiency, where the plane is tangent to
the paraboloid. Thus impedance and power relationships that are far from
obvious may be visualized easily.
7.3,1. Output Power Parabolaid. Power transfer from a complex source to a
complex load was considered in Section 3.2.3. The load power. normalized to
the source power available, was expressed in (3.47):
This has the form of a parabola,
y=l-x',
(7.44)
(7.45)
where x is the radius corresponding to a constant reflection magnitude, i.e.,
constant normalized output power.
To extend these results to the output port of a linear two-port network,
consider the model in Figure 7.10; it is consistent with the defining admittance
parameter equations, (3.79) and (3.80). in Section 3.4.3. For constant VI = I.
an equivalent Norton source at output port 2-2' has the available power
p
_ IY'II'
,0- 4g
22
' (7.46)
, I, 2 I,
Figure 7.tO.
"
An admittance parameter model consistent with defining equations.
2'