Maximum-Power-Transfer Power Coordinates for Fully Coupled Multiport Thévenin Sources
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 $R_s=\mathrm{Herm}\{Z_s\}\succ\mathbf{0}$, completing the square in accepted power identifies the available-power current and motivates the coordinates $\mathbf{a}=\frac12R_s^{-1/2}(\mathbf{V}+Z_s\mathbf{I})$, $\mathbf{b}=\frac12R_s^{-1/2}(\mathbf{V}-Z_s^H\mathbf{I})$. They satisfy $\|\mathbf{a}\|_2^2-\|\mathbf{b}\|_2^2=\Re\{\mathbf{I}^H\mathbf{V}\}$ 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 $P_{acc}>0$, $P_{rad}/P_{av}=η_{rad}(1-\mathrm{TARC}^2)$, 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.