# Stable Dissipative Linear Stationary Dynamical Scattering Systems

• D. Z. Arov
• J. Rovnyak
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)

## Abstract

In the theory of passive linear electrical networks, the Darlington method is well known as a realization of a finite ideal passive 1-port with losses via a finite ideal passive lossless 2-port closed by one resistance [9]. The reflection coefficient Θ of such a 1-port is an element of the scattering matrix $$\tilde \Theta$$ of a corresponding loss-less 2-port; the lossless behavior is indicated in the property that $$\tilde \Theta$$ has unitary values on the boundary of the physical domain (in the right or upper half-plane, or inside the unit disk). The consideration of scattering matrices allowed Belevich to generalize Darlington’s result on finite ideal n-ports with losses [16]. Darlington himself did not consider Θ and $$\tilde \Theta$$ but other frequency characteristics: the impedance Zof 1-ports and the transmission matrix Ã of 2-ports (Z and Ã have simple representations by means of Θ and $$\tilde \Theta$$). In this way, the Darlington result was generalized to finite ideal n-ports with losses by V.P. Potapov [14] and his student E.Ya. Melamud [12].

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