On Matrices of Residues of the Impedance or Admittance Matrices ofnPorts

In the paper some properties of matrices of residues for the open-circuit impedance or the short-circuit admittance representation of
n
-ports are discussed. It is shown that at a simple pole, located wherever in the complex plane, the matrix of residues of the impedance or admittance matrix is, in general, highly degenerate. The rank of the matrix of residues of the impedance matrix cannot exceed the nullity of the numerator of the admittance matrix, and vice versa. The above proposition turns out to be a special case of a more general theorem for multiple poles. At a multiple pole, instead of the matrix of residues, the matrix of the first coefficients in the Laurent expansion about this singular point is to be considered. At poles on the imaginary axis where, according to Cauer, the matrix of residues is positive definite or semidefinite, the application of the above discussion shows that such a matrix is, in general, semidefinite (and of unit rank) and it may be positive definite only in very special cases.