On linear multivariable and multiloop feedback networks

The authors present general, compact, and efficient formulas to reduce computational time in calculating the general null return-ratio and return-difference matrices of a linear multiinput multioutput and multiloop feedback networks 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, where the transfer impedance matrix need not be diagonal. The authors assume that the networks under consideration comprise only elements such as resistors, inductors, capacitors and voltage-controlled current sources. The authors consider the general situation where the return-difference and null return-difference matrices are computed for a general reference matrix of residues instead of the zero matrix. The physical significance to these residue effects is that in computing the sensitivity function and in the measurement of the return ratio for practical feedback amplifiers, the general reference values correspond to the situation where the feedback amplifier under study is made partially active rather than completely dead, as in the original interpretations of the return-ratio and return-difference matrices for the zero reference value