Phase Preservation of $N$-Port Networks Under General Connections

This study first introduces the frequency-wise phases of $n$-port linear time-invariant networks based on recently defined phases of complex matrices. Such a phase characterization can be utilized to quantify capacitive, inductive, and passive behaviors of $n$-port networks, as well as to relate to the power factor of the networks. Further, a class of matrix operations induced by fairly common $n$-port network connections is examined. The intrinsic phase properties of networks under such connections are preserved. Concretely, a scalable phase-preserving criterion is proposed, which involves only the phase properties of individual subnetworks, under the matrix operations featured by connections. This criterion ensures that the phase range of the integrated network can be verified effectively and that the scalability of the analyses can be maintained. In addition, the inverse operations of the considered connections, that is, network subtractions with correspondences are examined. With the known phase ranges of the integrated network and one of its subnetworks, the maximal allowable phase range of the remaining subnetwork can also be determined explicitly in a unified form for all types of subtractions. Finally, we extend the phase-preserving properties from the aforementioned connections to more general matrix operations defined using a certain indefinite inner product.