## 1 Introduction  
The ElectroMagnetic Launch (EML) technology uses electric propulsion to accelerate objects at high speeds. Because of its superior performance it is substituting several launch systems. Coil and rail launchers are the most used alternative solutions [1, 2]. Induction launchers are substantially linear tubular motors, usually air cored. They consist of a barrel formed by an array of (stator) coils and a conductive cylinder moving inside them. Induction launchers are operated as travelling wave induction launchers or as pulsed induction launchers. In the first operation mode, the stator coils are grouped in sections that are energized in a polyphase fashion in order to create a traveling wave of flux density in the region occupied by the sleeve. In the second one the stator coils are fed in sequence by a set of capacitor driven circuit [3, 4, 5, 6, 7]. The Electro-Magnetic Aircraft Launch System (EMALS) is another important example of application of the electromagnetic launch technology. It has been introduced in substitution of the steam catapults for the take-off of airplanes from the new class of carriers of the US Navy [8; 9; 10; 11]. With respect to the steam catapult EMALS is able to produce a smooth and controllable acceleration profile with a consequent reduction of the stress on the aircrafts. Moreover it is able to accelerate heavier aircrafts with reduced weight, cost and maintenance requirements. EMALS has been recently proposed for civil aircrafts [12]. Amibitious programs for space application of electromagnetic launchers are under investigation [13; 14]. A rail launcher is constituted by two conductive rails with a conductive armature (slab or c-shaped) free to slide inside them. At the beginning of the launch the armature is located near the breech of the launcher where a feeding generator is connected between the rails. The current flowing in the rails (and in the armature) produces a flux density distribution in correspondence of the armature, where the interaction with its current produces a thrust force that accelerates the armature. The main drawbacks affecting the rail launchers are a consequence of the Velocity Skin Effect (VSE) which is caused by the limited diffusion rate of the current in the rails as the armature moves; VES produces a concentration of the current in the rear portion of the armature near the rails [15; 16; 17; 18]. The importance of VSE increases with the speed and it is one of the causes which may prevent the use of rail launchers at very high speed. The availability of numerical tools for the investigation of VSE and for the design of countermeasures to limit its effects on the launcher performance are of paramount importance [17; 19; 20]. When considering a solid armature rail launcher, the choice of an air-core compensated pulsed alternator (compulsator) as the feeding device seems to be one of the most promising technology [21]. The absence of ferromagnetic materials allows achieving a very low value of internal inductances. The addition of compensating windings or conductive shields further reduce the internal inductance, so increasing the peak value of the output pulsed current. Moreover, by proper positioning of compensating components, it is possible to shape the current pulse to improve the performance of the launchers, both in terms of muzzle speed and efficiency. As reported in the scientific literature, the maximum speed of an air-core rotor can reach higher values than those in an iron-core one, increasing the stored energy [22; 23]. Many papers, based on analytical or numerical models, have been published in the past years to investigate the performance of the air-core compulsator [24; 25; 26]. However, the majority of these studies are focused on the performance of the compulsator as a stand-alone device and adopt a simple time varying equivalent circuit to model the rail launcher. Similarly happens for rail launchers, where often the waveforms produced by the feeding devices are assigned, especially when the rotating machines are considered.