Maximum-Power-Transfer Power Coordinates for Fully Coupled Multiport Thévenin Sources

Abstract

A power-normalized scattering representation is developed at a fixed frequency for a passive linear time-invariant multiport load driven by a fully coupled multiport Thévenin source. The source is obtained by reducing, at the load reference planes, an independent-source network whose suppressed internal impedance is passive, together with an intervening passive matching network. No diagonal-reference, uncoupled-source-channel, reciprocity, or commutation assumption is required. With $R_s=\mathrm{Herm}\{Z_s\}\succ 0$, completing the square in accepted power identifies the available-power current and motivates the coordinates $\mathbf{a}=\frac{1}{2}{R}_s^{-1/2}(\mathbf{V}+Z_s\mathbf{I})$, $\mathbf{b}=\frac{1}{2}{R}_s^{-1/2}(\mathbf{V}-Z_s^H\mathbf{I})$. They satisfy $\Vert \mathbf{a}{\Vert }_2^2-\Vert \mathbf{b}{\Vert }_2^2=\Re \{{\mathbf{I}}^H\mathbf{V}\}$ and yield $\mathbf{S}=R_s^{-1/2}(Z_{load}-Z_s^H)(Z_{load}+Z_s{)}^{-1}{R}_s^{1/2}$. An exact operator identity establishes passivity--contractivity equivalence and gives excitation-specific, reachable-subspace, and complete conjugate-matching conditions. On the physically reachable incident subspace, singular values characterize the best- and worst-case source-normalized port-reflection TARC, while a restricted Frobenius norm gives the basis-averaged squared TARC. Equal-magnitude phase-only control is formulated separately as a constant-modulus problem, with generator-side constraints mapped through the coupled source network before power normalization. For antenna loads and excitations with $P_{acc}>0$, $P_{rad}/P_{av}={\eta }_{rad}(1-{\mathrm{TARC}}^2)$, so terminal scattering data alone do not determine radiation efficiency. When $R_s$ is singular, finite available power exists exactly for $\mathbf{E}\in \mathrm{range}(R_s)$, and the construction applies on the positive-resistance support.

19 pages, 1 figure