Stability and Power-Gain Invariants of Linear Twoports

It is shown that the stability of a linear twoport is invariant under arbitrary lossless terminations, under interchange of input and output, and under "immittance substitution," a transformation group involving the arbitrary interchanging of impedance and admittance formulations at both ports. The quantity
k = \frac {2 Re (\gamma_{11}) Re (\gamma_{22}) - Re(\gamma_{12} \gamma_{21})} {|\gamme_{12} \gamma_{21}|}
(where the
\gamma
may be any of the conventional immittance
z, y,
or hybrid
h, g
matrix parameters) is the simplest invariant under these transformations, and describes uniquely the degree of stability, provided
Re(\gamma_{11}), Re(\gamma_{22}) \geq 0;
the larger
k
is, the greater the stability, and in particular
k = 1
defines the boundary between unconditional and conditional stability. The quantity
k
is thus the basic invariant stability factor. Its definition is also extended to include the effect of terminating immittances, which may be padding resistances or source and load immittances, or both. Certain power-gain functions, including the maximum available power gain, are shown to be invariant under immittance substitution, and
k
is identified as a function of ratios between them, where they exist. This provides a fundamental way of determining k, apart from calculating it from matrix parameters, and indicates that it is a measure of an inherent physical property.