The Calculation of Energy Flow using the Laplace Transformation

Energy flow has usually been calculated by integrating the product of two functions, one of which can often be regarded as an applied force and the other as a resultant response. When there is no interest in the explicit nature of this response, its determination in order to obtain the energy flow is an undesired labour. In the paper it is shown that the Laplace-Parseval integral can be usefully employed to determine energy flow directly from a knowledge of the Laplace transforms of the applied force and response. This procedure involves no more work than that required to invert the transform of the response, and the saving of effort is thus equal to that involved in the integration of the product. The method is shown to be readily applicable to both transient and steady states, and in all but the most elementary problems there is a considerable reduction of the risk of errors arising. Moreover, the method is shown to lead directly to the most simple form of solution, which has been attainable hitherto only after algebraic reduction. By way of illustration, the classical problem of eddy-current losses in linear sheet conductors is solved in general for arbitrary excitations. The method is shown to hold most advantage when the applied force possesses a number of discontinuities.