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Structure of quasi-primitive multiports

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摘要

A quasi-primitive n-port is described by an admittance matrix of the form Y(p)= p{sup}-1 A+B+pC in which A, B, C, are real, symmetric, semidefinite n×n matrices. Any m-port, m＜n, derived by ignoring some chosen (m-n) ports of the n-portdescribed by Y(·) is a descendant of Y(·).Any such descendant is described by its own admittance matrix Y'(·)called then a descendant of Y(·). This note presents two results, the first a "canonical" form into which any given quasiprimitive Y(·) canbe put by a transformation of coordinates. Specifically, given Y(·), quasi-primitive and of size n×n, there exists a real, invertible, n×n matrix Q such that Q{sup}＊Y(·)Q is of block-diagonal form bldiag {Y{sub}SP(p),Y{sub}ND(p},Y{sub}TI(p)} in which the blocks are quasi-primitive and describe respectively an n{sub}-1-port, an n{sub}2-port, a 2n{sub}3-port, separate networks, each of which, if not of size zero, exhibits distinctive characteristic properties. The second result, based on this structure, is a property of a given m×m positive-real matrix-valued function Y'(·) necessary that Y'(·) be a descendant of some quasi-primitive matrix Y(·).

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