Additional findings on S-parameter bounds valid for unconditionally stable N-ports

In a recent paper, it has been shown that, if an N-port network fulfills the condition of (geometrical) unconditional stability at a given frequency, then its scattering parameters will also necessarily satisfy N easily computable bounds, one per port. In order to complete that picture, this contribution investigates whether a tighter bound can be obtained by combining the $N$ bounds into just one. The answer is in general negative, except that the 3-port case does indeed exhibit a peculiar behavior: this can be exploited to reduce the upper bound when the diagonal elements of the scattering matrix are limited in magnitude up to some $\alpha$, and in particular for $\alpha =0$ (simultaneous conjugate match).