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238 Linear Amplifier Design Tools .
real normalizing impedances. It was remarked that left-half-plane impedances
are represented in the reflection plane region outside the unit circle and thus
off the Smith chart. However, plotting 1/p* for these impedances, where
Ipl> I, enables the use of the Smith chart in a fairly normal way. The
resistance loci must be read as the negative of their usual values and the
reactance loci are read normally.
7.2. Impedance Mapping
Impedance mapping is a method that allows a peek into the complex plane
associated with any Z, Y, or S response parameter. What one is able to "see"
there is a small, rotated, generalized Smith chart representing the entire
impedance, admittance, or scattering plane of any network branch. Even more
generally, the impedance-mapping formulation enables the restatement of any
bilinear function into a form having a complex translation constant and a
complex factor that scales and rotates the generalized Smith chart's unique
bilinear form (7.16). Impedance mapping is very valuable for visualization,
analysis, and computation.
In this section, a linear three-port network will be characterized by its
scattering parameters and one port terminated by a fixed reflection coefficient.
The equivalent two-port parameters will be derived. This has value in
ladder network analysis when a terminated three-port circulator appears in
cascade. An HP-67/97 program is provided for this transformation. More
generally, this result proves the important bilinear theorem, which 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.
For example, response S12 must be a bilinear function of any branch
impedance, say Zb' This has many practical applications in neutralization,
oscillator, filter, and amplifier design.
Finally, the impedance-mapping relationships will be derived, and a handheld
computer program will be furnished. Many examples will be provided to
illustrate these principles and applications.
7.1./. Three-Port to Two-Port Conversion. Two-port scattering parameters
were considered in Section 4.5.2; the defining system of linear equations was
given in (4.46) and (4.47). In general, such systems for any number of ports
may be described in matrix notation as
b=Sa.
(7.20)
This notation for three-port networks stands for
b l = sllal + S12a 2 + S13a 3,
b 2 = S21 a l + S22a 2 + S23a 3, (7.21 )
b 3 = S31 a l + S32a 2 t S33a 3 .
Three-port scattering parameters will be lower-case sij' and two-port parame-