[en] Highlights: • In this article, for the eddy current damper attached to the HTSC, we • quantitatively investigated the damping coefficient c, damping ratio, Joule heating of the copper damper, and the vibration frequency of the PM as well. • presented four different arrangements of the copper damper, and comparatively studied their damping effects and Joule heating, and finally proposed the most advisable arrangement. - Abstract: Passive superconducting levitation systems consisting of a high temperature superconductor (HTSC) and a permanent magnet (PM) have demonstrated several fascinating applications such as the maglev system, flywheel energy storage. Generally, for the HTSC–PM levitation system, the HTSC with higher critical current density J_c can obtain larger magnetic force to make the PM levitate over the HTSC (or suspended below the HTSC), however, the process of the vibration of the levitated PM, provides very limited inherent damping (essentially hysteresis). To improve the dynamic stability of the levitated PM, eddy damping of additional conductors can be considered as the most simple and effective approach. In this article, for the HTSC–PM levitation system with an additional copper damper attached to the HTSC, we numerically and comprehensively investigated the damping coefficient c, damping ratio, Joule heating of the copper damper, and the vibration frequency of the PM as well. Furthermore, we comparatively studied four different arrangements of the copper damper, on the comprehensive analyzed the damping effect, efficiency (defined by c/V_C_u, in which V_C_u is the volume of the damper) and Joule heating, and finally presented the most advisable arrangement.