## 3 Weak Interaction Analysis  
In this section the analysis based on the \"weak interaction\" between rail launcher and compulsator will be critically reviewed and the difficulties of the equivalent circuit extraction will be discussed. Figure 3 shows the active parts of a single-phase, two-pole compulsator with selective passive compensation provided by a discontinuous conductive shield. The inner part of the device consists of two stationary field coils which produce a magnetic flux density distribution whose axis is in the vertical direction (Figure 3b). The two series connected armature windings are located on the rotor which is the outer part of the machine. Figure 3c shows the stationary compensating shields that are in the central part of the machine. Figure 3a shows a 3D view of the device. A more detailed description is reported in [29]. When using a compulsator to feed a rail launcher, its rotor has to be preliminarily driven to a proper angular speed at no load conditions; part of the stored kinetic energy of the rotor will be delivered to the railgun armature. At the firing instant, the armature windings of the compulsator are connected to the rails of the launcher. During the launch, the rail armature accelerates and at the same time, the rotating part of the compulsator decreases its angular speed. The rates of change of the speeds of both the moving parts (and consequently their positions) are not known a-priori and substantially depend on the delivered current. Since the simultaneous 3D analysis of the two electromechanical devices can be very time consuming, a common practice is to substitute one or both the device with equivalent circuits. This procedure may lead to significant error in the current flowing in the devices because of the difficulties in the determination of the topology and of the parameter extraction of the equivalent circuit. Referring to the compulsator, when looking for a lumped equivalent circuit, we have to distinguish between the two compensation techniques usually adopted. When compensation is achieved by the use of shorted discrete coils, we can consider the self and mutual inductances of all windings (field, armature and compensating ones). Considering that the values of the mutual inductances depend on relative angular positions only, the lumped equivalent circuit of the compulsator is able to give accurate results; its building is a matter of evaluation (experimental or by computations) of self and mutual coefficients and EN4 EM can be used as an extraction tool. Things go in a different way when non-uniform (discontinuous) compensating shields are used. In these devices, the extraction of the values of the equivalent internal inductance \(L\) is not as straightforward as with shorted discrete coils, since it is function of the position of the rotor, as well as of the speed of the rotor, which is not known and varies during the system operation. Figure 3: An example of the modeling of a compulsator by EN4 EM. Snapshot of (**a**) the 3D view; (**b**) cross section of the machine; (**c**) discretization of the compensating shield. (Reproduced with permission from [29], IEEE, 2017). In fact, the speed of the rotor imposes the frequency of the electrical quantities in the shield, which in turn determines the path and the amplitude of the induced currents and their shielding effects which contribute to the internal equivalent inductance. Figure 4 shows an example of eddy currents distribution on one of the discontinuous compensating shields of the compulsator above described whose operating conditions are reported in [29]. In particular the compulsator was loaded with a simple lumped R-L equivalent circuit representing the rail launcher. The rectangle in Figure 4 represents the flattened cylindrical surface shown Figure 3c. The equivalent internal inductance of the compulsator, as well as the electromotive force at the terminals of the machine, are functions of the distribution of the currents on the shields.