Updatable Closed-Form Evaluation of Arbitrarily Complex Multi-Port Network Connections

The design of large complex wave systems (filters, networks, vacuum-electronic devices, metamaterials, smart radio environments, etc.) requires repeated evaluations of the scattering parameters resulting from complex connections between constituent subsystems. Instead of starting each new evaluation from scratch, we propose a computationally efficient method that updates the outcomes of previous evaluations using the Woodbury matrix identity. To enable this method, we begin by identifying a closed-form approach capable of evaluating arbitrarily complex connection schemes of multi-port networks. We pedagogically present unified equivalence principles for interpretations of system connections, as well as techniques to reduce the computational burden of the closed-form approach using these equivalence principles. Along the way, we also achieve the closed-form retrieval of the power waves traveling through connected ports. We illustrate our techniques considering a complex meta-network involving serial, parallel and cyclic connections between multi-port subsystems. We further validate all results with physics-compliant calculations considering graph-based subsystems, and we conduct exhaustive statistical analyses of computational benefits originating from the reducibility and updatability enabled by our approach. Finally, we find that working with scattering parameters (as opposed to impedance or admittance parameters) presents a fundamental advantage regarding an important class of connection schemes whose closed-form analysis requires the treatment of some connections as delayless, lossless, reflectionless and reciprocal two-port scattering systems. We expect our results to benefit the design (and characterization) of large composite (reconfigurable) wave systems.