**2. Analytical methods**

The magnetic top is composed of a couple of ring-shaped permanent magnets magnetised in the axial direction, as shown in Figure 1. The magnetic top, equipped with a smaller ring-shaped permanent magnet (a rotor magnet), can be levitated in the magnetic field generated by the larger ring-shaped permanent magnet (a stator magnet) situated at its base, if it can maintain its rotation within a certain speed range. The levitating height is determined by the shapes and magneto-motive forces of the permanent magnets. The authors propose two types of analytical methods: (1) a quasi-three-dimensional analysis to investigate the principle of levitation and the design parameters of the permanent magnets and (2) a three-dimensional dynamic analysis to simulate the behaviour of the levitating magnetic top. The ring-shaped permanent magnet is approximated to the equivalent coil currents model in both the analytical methods.
