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
Tavşanoğlu, Ahmet Vedat
MetadataTüm öğe kaydını göster
This paper considers the task of constructing an (MxN+1)-node rectangular planar resistive grid as: first forming two (MxN+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 shown that the nodal conductance matrices GH and GV of the two sub-grids can be expressed as the Kronecker products GH = I-M circle times G(N), G(V) = G(M)circle times I-N, and G of the resultant planar grid as the Kronecker sum G = G(N circle plus) G(M), where G(M) and I-M are, respectively, the nodal conductance matrix of a linear resistive grid and the identity matrix, both of size M. Moreover, since the analytical expressions for the eigenvalues and eigenvectors of G(M) - which is a symmetric tridiagonal matrix- are well known, this approach enables the derivation of the analytical expressions of the eigenvalues and eigenvectors of G(H), G(V) and G in terms of those of G(M) and G(N), 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.