*4.3. Losses in the Reaction Rail*

In order to determine the transverse effects of the current flowing in the reaction rail, another numerical FE model should be applied. This can be done using the electric vector potential *T*, defined by the formula *rotT* = *J*, where *J* denotes the current density vector [74,75]. The differential equation for the electrical vector potential can be written as follows (movement only in *x*-direction is allowed):

$$\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} = \sigma (\frac{\partial B}{\partial t} + v\_x \frac{\partial B}{\partial x}) \tag{2}$$

Solving Equation (2) with different values of σ, *vx*, and *f*, it is possible to calculate the current density distribution (and other important parameters of the LIM, such as power losses and forces) for any combination of these parameters. Figure 13 shows an example of the current density distribution in the aluminum rail of the LIM for different slip values of constant frequency.

**Figure 13.** Current density distribution in the rail of the LIM with constant frequency and different slip values [29].

The electric vector potential method can also be applied for the calculation of eddy current distribution in the copper sheet of the rotor due to skewed armature slots (Figure 14a) and for the analysis of influence of the rotor slits on the eddy current distribution in the rotor of rotating induction machines (Figure 14b).

**Figure 14.** Eddy current distribution in the copper sheet of the rotor due to skewed armature slots (**a**). Influence of the rotor slits on eddy current distribution in the rotor (**b**) [89,90].

The method presented here has a broader meaning. It is a practical tool that enables the analysis of the power losses in the LIM reaction rail and their minimization. Similar results have also been presented in [55].
