Instantaneous Power Balance in Finite-Element Simulation of Electrical Machines

Conservation of power in time-stepping finite-element (FE) simulation of electrical machines is studied. We propose a method for accurately obtaining the instantaneous time derivative of the FE solution, from which the instantaneous eddy-current losses and the rate-of-change of the magnetic field energy are calculated. The method is shown to be consistent with different time-integration schemes, unlike the typically used backward-difference (BWD) approximation, which is only accurate if the BWD method is also used for the time integration. We first formulate the FE equations for a locked-rotor induction machine as a differential-algebraic equation (DAE) system. An approach called the collocation method is then used to derive the BWD, trapezoidal (TR), and implicit midpoint integration rules in order to show how these methods approximate the solution in time. We then differentiate the constraint equations of the DAE to form a system from which the time derivative of the solution can be solved. The obtained derivative is shown to satisfy the power balance exactly in the collocation points. In case of the TR rule, the losses calculated with the proposed method are shown to be less sensitive to the time-step length than ones obtained with the BWD approximation for the time derivatives. The collocation approach also allows studying the power balance continuously during the time step.