The value of the field from the air-gap is given (demonstration of the answer to the question 2) by

Considering, it results .

According to the curve of the material's magnetization, it results . This corresponds to a relative permeability of the material:

Considering this value of the relative permeability we calculate to which it corresponds the value of induction and consequently

Through successive iterations we find a converging point for:

and

Regardless of of the relay armature when , the electromagnetic torque is contained within the maximum

previously calculated for and maximum value

The corresponding curves are drawn in figure 7.

Figure 7: The maximum and minimal values of the torque depending on the position (assuming the saturation of the magnetic material)

We can observe that, round position, the torque produced by the engine decreases to . So the saturation effect determines the reduction with over 50% of the electromagnetic torque (compared to the liniary, unsaturated case).

Calculating for each position the value of the relative permeability of the magnetic environment, we can draw the curve (figure 8).

Figure 8: The torque dependency vs. the position (assuming the saturation of the magnetic material)

The calculation considers as negligible

• the stray flux

• the local effects of saturation.

The finite elements modelling technique allows the numerical integration of the local ecuations of magnetic field (Maxwell's equations) in any point from space, which allows a more precise estimation of the torque depending on position.

Figure 10 (obtained with the help of the FLUX2D programme, developed by Cedrat company) emphasizes the idea that, when the two armatures are aligned, the induction in the areas close to the air-gap can localy reach over 0.7T (figure 9).

In this case, the magnetic material is completely saturated. To reach this level of induction, the H_f field must localy exceed the value of 100.000 A/m.

The relative permeability of the magnetic environment will not exceed in these areas the m_r = 6 value. These effects of the local saturation, explains largely, why the torque calculated through the finite elements technique, is even smaller than the one obtained through analytical calculation (figure 11).

Forwards, the saturation phenomenon contributes to the increasing of the stray flux, respectively to the magnetic flux produced by reels, which does not close through the relay armature. (figure 12, obtained with the help of the FLUX2D programme, developed by Cedrat company. ).