Magnetic levitation (maglev) is used to suspend and propel trains without contact with a rail, and many cases, used to support rotating bearings for high-performance machines in order to eliminate friction and the need for lubrication. In the magnetic levitation system shown, the current-carrying coil produces an electromagnet that pro- vides an attraction force Fem on the metal ball: ein (1) Fem where K is a "force constant" (units of N-m/A) that depends on the number of coil wraps, material properties of the electromagnetic core, and geometry of the electro- magnet. Position of the ball is measured upward from a fixed at-rest position and distance d is a constant equal to the nominal air-gap distance from the electromagnet tip and the ball for a nominal coil current. R www KEI (d- x) Nominal air gap, d sa I (for x = 0) Metal ball, m Electromagnet Coil, L Derive the complete mathematical model of the magnetic levitation system where voltage source ein (t) is the input variable.