A method for analyzing a magnetic damper (eddy current brake) consisting of a cylindrical magnet and a plate conductor of arbitrary shape is considered. Since the magnetic flux is a function of the position, the analytical solution to obtain the eddy current, braking force, and damping coefficient is obtained by dividing the magnetic flux into the narrow circular bands, and the unit step function is applied to solve the differential equation of the electromagnetic fields. The boundary condition of the plate conductor of arbitrary shape is satisfied directly by making use of the Fourier expansion collocation method. Numerical calculations have been carried out for the conductor of rectangular plates, circular plates with eccentric fluxes. The theoretical results are in very good agreement with the experimental ones.