Analysis and computation of feedback matrices in multivariable and multiloop feedback networks

Compact formulas are presented expressing various return ratio and return difference matrices and the null return ratio matrix of a linear multivariable and multiloop feedback network in terms of the first- and second-order cofactors and their partial derivatives as well as the third-order cofactors of the elements of the indefinite-admittance matrix (IAM), using the matrix signal-flow graph (MSFG) as a tool. This requires that all the controlled sources of the network be first converted to voltage-controlled current sources. However, the excitations can either be the current sources, voltage sources, or any combination of these sources, but the current sources are preferred. The formulas are useful in computing the feedback matrices in that they do not require any matrix inversion in computing the return ratio matrix and require only one inversion in computing the null return radio matrix. In addition, they are suitable for symbolical analysis and are especially convenient when the direct transmission matrix is diagonal