A Bound on the Scattering Parameters of Unconditionally Stable N-Ports

Unconditionally stable 2-ports exhibit the peculiar property of mapping regions in the passive termination plane into smaller regions in the input plane. Such a ‘contractive’ behavior is observed also when the number of ports, $N$ , is larger than two, provided the notion of output region is appropriately adjusted. Based on this premise, it is shown that the magnitudes of the scattering parameters of an unconditionally stable $N$ -port are bounded by a simple inequality. This necessary condition generalizes similar conditions which are already known to be valid for passive, reciprocal $N$ -ports, on the one hand, and for generic unconditionally stable 2-ports, on the other.

I. Introduction

The theory of linear circuits provides well known boundaries on the magnitudes of the scattering parameters of passive, reciprocal circuits. In particular, when such circuits are strictly passive, they must obey the following relationship:

$$\begin{equation*} \sum _{j=1}^{N} \left |{S_{ij}S_{ji}}\right | < 1 \, ~\forall \, i \tag{1}\end{equation*}$$ where is the number of ports [1], [2], [3]. If the passivity condition is relaxed to also include lossless networks, the < sign is replaced by a sign.