On Brune´s Tests

Brune´s tests are well known in circuit theory. They are two simple voltage tests used to ascertain whether the overall component stemming from the parallel-parallel, series-series, series-parallel, or parallel-series connection of two four-terminal components has a two-port matrix that is equal to the sum of the appropriate individual two-port matrices. What is less known is that Brune´s tests only are necessary and sufficient for reciprocal components and that all the extensions that are available in literature to also cover the nonreciprocal components are inaccurate. In this brief, a rigorous and thorough unified analysis of the four connections is carried out. It provides the complete characterization of the overall component under a two-port operation in the general case and, most importantly, the exact parameter conditions and voltage/current tests for the validity of the matrix sum rule.