Skew symmetry and orthogonality in the equivalent representation problem of a time-varying multiport inductor

This paper considers the fundamental problem of passive multidimensional Kirchhoff networks for linear time-varying systems that are suitable for wave digital filter discretization. An explicit solution, subject to the validity of a commutativity condition, is given in the linear time-varying case for the feasibility of representation of the coupled inductor, the crucial dynamic multiport in the network, in two equivalent forms so that the property of losslessness of the coupled inductor, and therefore, passivity of the entire network, is assured by the nonnegative definiteness of the inductance matrix for all space and time variables. The commutativity condition is expressed in an equivalent form that requires the product of two skew-symmetric matrices to be symmetric. An isomorphism is developed and proved between the spaces of skew-symmetric and orthogonal matrices of a common order. The feasibility of generalization of these results to the case of nonlinear current-controlled coupled inductor matrix is briefly explored and illustrative examples are provided throughout to facilitate comprehension of the concepts.