Generalised cascade sum of matrices associated with active RLC multiports through generalised inverses

The cascade connection of two active RLC multi-ports described by the terminal impedance matrices {not necessarily nonsingular or h.s.d. (hermitian positive semidefinite)} is used to define an operation, called the generalised cascade sum. In a previous work, the cascade sum obtained by Trapp and Duffin utilises the h.s.d. impedance matrices as well as the pseudoinverse (Moore-Penrose generalised inverse) as a tool. The cascade sum of matrices, considered in this paper, corresponds to those networks which may contain not only passive resistors but also ideal transformers and active RLCs. It is shown that the method described for generalised cascade sum of matrices is independent of the choice of generalised inverses. In fact, the terminal impedance matrices need not be h.s.d. Further, in a more general case, the impedance matrices of the multiports may not even possess the property of hermitian semidefiniteness, yet the explicit formula of the generalised cascade sum of matrices can still be obtained through a set of necessary and sufficient conditions given in the paper which involves any one of the generalised inverses.