A frequency-domain solution for the motion of levitated conductors

Local coordinate systems are attached to the conducting bodies in motion which allows the usage of the eddy-current integral equation to determine the electromagnetic field. The space discretization is performed only for the conducting bodies and remains unchanged in time. An additional term, similar to the “motional” electromotive force in Faraday´s law, is introduced in the integral equation. If the imposed currents are sinusoidal with time, a phasor representation is employed for a frequency-domain analysis based on the eddy-current integral equation. Since the “electrical” period is much smaller then the “mechanical” period of the body vibration, the time average force over an “electrical” period is used in the equation of motion. This allows for the solution of the equation of motion to be obtained by employing a time step which is greater than the “electrical” period. Thus, the efficiency of the method presented is much greater than the efficiency of the time-domain methods, where the time step is chosen to be a small fraction of the “electrical” period. This is more so at higher frequencies, which are used for more efficient levitation devices.