The return-difference matrix in linear networks

Bode´s concept of the ¿return difference¿ is generalized by replacing the quantities concerned by matrices. A formula for the ¿fractionated gain¿ is derived, which can be considered as a generalization of Thévenin´s theorem in three respects. First, it is applicable to active circuits; secondly, several terminals are opened simultaneously; and thirdly, the voltages or currents or both, about which a statement is made, need not be those at the opened terminals. Formulae based on the concept of the return-difference matrix are derived for the stability of a circuit, the sensitivity of its output to the imperfections of some of its unilateral or bilateral elements, and for its input and output impedances. All derivations are based on the superposition and compensation theorems. The concept of the return-difference matrix is particularly useful in the analysis of multiple-loop feedback circuits.