On Thevenin-Norton and Maximum power transfer theorems

TL;DR

This paper develops a general method for analyzing the behavior of complex multiports with arbitrary port equations and introduces a broad version of the maximum power transfer theorem based on stationarity conditions when terminating the network with its adjoint.

Contribution

It presents a novel approach for computing multiport behavior with general port equations and generalizes the maximum power transfer theorem without requiring hybrid matrices.

Findings

Method works for linear multiports with arbitrary port equations.

Stationarity of power transfer occurs when the multiport is terminated by its adjoint.

The most general version of the maximum power transfer theorem is established.

Abstract

In this paper we show how to compute port behaviour of multiports which have port equations of the general form $B v_P - Q i_P = s,$ which cannot even be put into the hybrid form, indeed may have number of equations ranging from $0$ to $2n,$ where $n$ is the number of ports. We do this through repeatedly solving with different source inputs, a larger network obtained by terminating the multiport by its adjoint through a gyrator. The method works for linear multiports which are consistent for arbitrary internal source values and further have the property that the port conditions uniquely determine internal conditions. We also present the most general version of maximum power transfer theorem possible. This version of the theorem states that `stationarity' (derivative zero condition) of power transfer occurs when the multiport is terminated by its adjoint, provided the resulting network has a solution. If this network does not have a solution there is no port condition for which stationarity holds. This theorem does not require that the multiport has a hybrid immittance matrix.