On the Solution of Networks by Means of the Equicofactor Matrix

In the solution of electrical networks, there arise matrices with the property that the sum of the elements of every row and of every column equals zero. On the node basis this is a direct consequence of Kirchhoff's current law coupled with the fact that the currents are invariant to a change of all node potentials by the same amount. As a consequence, all the first cofactors associated with determinants of such matrices are equal. The authors have named all such matrices equicofactor matrices and have based a general discussion of the solution of networks on these matrices. A new sign notation is introduced and problems of admittance-impedance conversion are treated. A proof is given of a theorem-called by the authors Jeans' theorem-which relates to the second cofactors associated with the determinant of the equicofactor matrix. This theorem is a consequence of the fact that, in the solution of networks, it is immaterial which node (mesh) is taken as reference and which equation is considered superfluous and suppressed from the given set, since the final answer must be the same. The theorem also shows that only (n-1)^2 coefficients associated with an n -node ( n -mesh) network are independent, regardless whether the network be described on an admittance or impedance basis. It is therefore concluded that there is perfect duality between the admittance and impedance description of networks, whatever their complexity.