Available energy of improper lossless LLTIP one-ports

A method for evaluating the available energy of a class of black-box circuit components is proposed. The class consists of those linear lumped time-invariant passive one-ports of any order that are improper and lossless. The approach requires the utilization of tools and techniques from the calculus of variations (Euler–Lagrange equations and natural transversality conditions) and involves rather demanding algebraic and integro-differential manipulations. However, the final formula for the available energy has the simple structure of a quadratic form in the one-port external state vector, relatively easy to use or to implement. The various steps of the approach developed are illustrated through examples. Furthermore, to evidence the difference between available energy and stored energy, two degenerate and one non-degenerate open- box one-ports are discussed in some detail.