Construction of the nodal conductance matrix of a planar resistive grid and derivation of the analytical expressions of its eigenvalues and eigenvectors using the Kronecker product and sum

Abstract

This paper considers the task of constructing an (M×A+1)-node rectangular planar resistive grid as: first forming two (M×A+1)-node planar sub-grids; one made up of M of (N+1)-node horizontal, and the other of N of (M+1)-node vertical linear resistive grids, then joining their corresponding nodes. By doing so it is sho wn that the nodal conductance matrices GH and GV of the two sub-grids can be expressed as the Kronecker products GH = Im ⊗ Gn, Gv = Gm ⊗ In, and G of the resultant planar grid as the Kronecker sum G = Gn ⊕ Gm, where Gm and Im are, respectively, the nodal conductance matrix of a linear resistive grid and the identity matrix, both of size M. Moreover, since the analytical expression s for the eigenvalues and eigenvectors of Gm — which is a symmetric tridiagonal matrix — are well known, this approach enables the derivation of the analytical expressions of the eigenvalues and eigenvectors of Gh, Gv and G in terms of those of Gm and Gn, thereby drastically simplifying their computation and rendering the use of any matrix-inversion-based method unnecessary in the solution of nodal equations of very large grids.