# Active Electromagnetic Clutch for Crankshaft Decoupling from a Belt Drive System
## 3 Electromagnetic Model  
The proposed actuator relies on an electromagnet to provide axial force and open the clutch mechanism. Due to the cylindrical shape of the assembly, an electromagnet with axial symmetry is the most suitable. This configuration guarantees a homogeneous axial force pull throughout the active surface of the electromagnet, thus providing robustness against centrifugal and belt forces acting on the pulley. Furthermore, the electromagnetic problem formulation can be simplified into two dimensions, as presented in Figure 2(a). The parts that constitute the mechanical structure of the actuator are manufactured from soft steel. They also guide the magnetic flux of the electromagnet. A U-shaped iron core is added to close the path of the magnetic circuit and to protect the electromagnetic coil. The thickness of this U-shaped iron core is limited to prevent mechanical interference with the crankcase and to ensure there is sufficient space to house the coil. This coil is a ring-shaped copper winding with 280 turns. The number of turns was chosen to comply with the 12-V battery standard for passenger cars. Relevant parameters of the electromagnetic clutch are summarized in Table 1.  
Figure 3: Finite element simulation of the electromagnet with an air gap of 2.5 mm and coil current of 7.2 A. (**a**) Two-dimensional geometry with axial symmetry. (**b**) Magnetic flux density norm distribution (color map), out-of-plane magnetic vector potential (contours). Figure 2: Side and isometric views of the clutch architecture. The pulley shape is exploited to provide an actuator with symmetry with respect to the rotation axis. The proposed electromagnetic geometry was simulated in COMSOL Multiphysics at different mover positions, leading to diverse air gap values measured between the mover and the fixed iron plate below it (from zero to 2.5 mm). In each of these positions, the coil was supplied with currents ranging from 0 to 10 A. The finite element model was meshed with 30,000 triangular elements, leading to a skewness-based average element quality of 0.89. Each model simulation takes approximately 30 s to compute. Figure 3b depicts the magnetic flux density distribution at an air gap of 2.5 mm and a supply current of 7.2 A. It is observed that the proposed U-shaped iron path provides strong saturation in some of its portions, as the saturation point of the iron starts at 1.3 T. Nevertheless, saturation will not lead to an efficiency problem in this device because it operates statically most of the time, i.e., supplied with a constant current. On the contrary, if the iron path is set to a larger thickness, the amount of copper will not be sufficient to provide the necessary attraction force. Figure 4a,b show the coil flux linkage and electromagnetic force generated, respectively, as a function of the air gap and coil current. These results were obtained with 1066 finite element runs within the input ranges of interest. Since the magnetic flux passes through the air gap twice, the air gap reluctance is involved in the flux path and thus the flux linkage varies not only with the coil current but also with the air gap. Saturation effects are advisable in the flux linkage above 3 A of supply current. The force shows the typical highly nonlinear behavior as a function of input variables. The data extracted from finite element simulations are useful to enrich a dynamic model of the actuator. By applying [PERSON]'s law to the electromagnet,  
\[v_{c}=\frac{\mathrm{d}}{\mathrm{d}t}\lambda(i_{c},g)+i_{c}R, \tag{1}\]  
in which \(\lambda\) is the flux linkage, \(v_{c}\) is the voltage applied to the electromagnet, and \(R\) is the coil resistance. Note that the flux linkage changes with both the current of the electromagnet \(i_{c}\) and the air gap \(g\): \(\lambda=f_{\lambda}(i_{c},g)\). Accordingly, the force produced by the electromagnet will have the same mapping: \(F=f_{F}(i_{c},g)\).