Chu Circuit Models as Minimum Energy Representations of Wave Impedances

The recoverable energy of spherical wave impedances and their circuit realizations are studied. Results show that the ladder networks proposed by Chu are minimum energy representations, with recoverable energy exactly equal to the stored energy. This equivalence is demonstrated analytically and studied in detail for a single low-order mode, then generalized for higher order modes. Transient calculations using heuristic recovery sources for the lowest-order mode provide a ground-truth verification. The demonstrated equivalence closes the open problem as to whether Chu circuits overestimate the energy stored by spherical wave impedances.