Power Gain in Feedback Amplifier

A linear three-terminal device
Z
is imbedded in a lossless passive network N and the properties of the complete system, as measured at two specified terminal pairs, are described by the open-circuit impedances
Z_{11}, Z_{12}, Z_{21},
and
Z_22
. A search for properties of
Z
which are invariant under the transformation N leads to the quantity
U = \frac{|Z_{21}-Z_{12}|^2}{4(R_{11}R_{22}-R_{12}R_{21})}
where
R_{jk}
is the real part of
Z_{jk}
. Quantity
U
is independent of the choice of
N
and is (consequently) invariant under permutations of the three terminals and also under replacement of the open-circuit impedances by short-circuit admittances. If
U
exceeds unity at a specified frequency, then
N
can always be chosen to make
R_{11}
and
R_{22}
positive and
Z_{12}
zero at that frequency. Quantity
U
is identifiable as the available power gain of the resulting unilateral structure. An arbitrary coupling network may be decomposed into a portion which accomplishes unilateralization and a remaining complementary portion which provides feedback around the unilateralized structure. Such decomposition brings some of the notions of elementary feedback theory to bear upon nonunilateral circuit analysis and offers a viewpoint from which signal flow and power flow are simply related.