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258 Linear Amplifier Design Tools
This section begins with the general relationship for converting from one
set of port normalizing impedances to another set. This will provide the basis
for almost all further development. Then stability will be restated in scattering
parameter terminology. The stability factor is equivalent to that previously
formulated in y parameters; here, stability circles will be located on input and
output reflection planes.
Several gain expressions previously defined will be put in scattering terminology,
and several more gains will be defined. In particular, loci of arbitrary
gain values will be defined for use when stability is conditional or unconditional.
A number of examples will be worked. However, no programs are provided
for this section. They are so common for every major personal computer,
desktop and hand-held, that there is little reason for duplication. The emphasis
here will be on the origin of the expressions that are commonly employed
in current scattering analysis of linear networks. Applications include oscillator
as well as amplifier design. Since there are a number of recurring complex
constants required throughout this development, they are collected in Appendix
E. The equations will be referenced as (E.xx), where xx is the equation
number.
7.4.1. Changing S·Parameter Port Normalization. Referring to Figure 7.17,
suppose that the scattering matrix S is defined for port 1, normalized to Z"
and port 2, normalized to Z,. Port normalization was described in Section
4.5.2. Even though the Z, constants may be complex, they are usually 50+jO
ohms, as obtained by automatic measuring equipment, and the following is
easier to understand in this context. Suppose that the port normalizing
iIl}pedances are to be changed from Zj to Zi, resulting in the new scattering
matrix S'. The new scattering matrix has been related to the original one by
'Kurokawa (1965); his derivation is too general for the present discussion. The
transfonnation is
(7.89)
Matrices A, R, and U (the unit matrix) are diagonal, and the * superscript is
the conjugate operator. The diagonal elements in matrix Rare
Zi-Zj
f i = Z'.+Z~' , ,
and the diagonal elements in matrix A are
(7.90)
I-r* ~
A, = 11- :;1 VI - Ir;/' , (7.91 )
-where i= 1,2. Clearly, (7.90) defines the port reflection coefficient r, of the
mew normalizing constant, commonly with respect to 50 ohms. Thus r, and r,
will be called the source and load coefficients, respectively.