Multiple-loop feedback circuits

A new concept, the null-return-difference matrix, is introduced. It is an extension of Tasny-Tschiassny´s return-difference-matrix concept and Truxal´s null-return-difference concept. In terms of the return-difference and null-return-difference matrices, two new theorems are formulated for multiple-loop feedback circuits. First, it is shown that in any such circuit the closed-loop gain is equal to the direct-signal transmission ratio multiplied by the ratio of the determinant of the null-return-difference matrix to the determinant of the return-difference matrix. Secondly, it is shown that the ratio of the driving-point impedance to the value it assumes when all the stages of a multiple-feedback circuit are put into their reference conditions is equal to the ratio of the determinant of the short-circuit return-difference matrix to the determinant of the open-circuit return-difference matrix.