3.1. Simulation and Experimental Results
In this study, neodymium iron boron permanent magnets were used for permanent magnet coupling. The specific performance parameters of the permanent magnet are listed in
Table 1. In addition, copper conductor tubes, aluminum frames, and inner and outer steel drums made of silicon steel with high magnetic permeability and low electrical conductivity were used.
The structural parameters of the permanent magnet coupling are listed in
Table 2.
The FEM can discretize various different solution domains and can flexibly use element meshes of different structural shapes for division. Therefore, for models composed of multiple linear or nonlinear materials exhibiting complex structures and boundary conditions, the FEM can be applied to obtain a relatively accurate numerical solution. The material of the permanent magnet used in permanent magnet coupling is anisotropic. In addition, multiple materials are connected in a straight or curved manner, and the eddy currents are unevenly distributed. Thus, the FEM is suitable for solving the three-dimensional (3D) transient field of permanent magnet coupling to obtain its transmission torque.
The simulation software ANSYS Maxwell was used in this study to establish the FEA model [
28]. The mathematical model of the magnetic vector
A in the solution domain
is as follows:
The following three types of boundary conditions were set: (a) inner surface of the model as
Sbc1; (b) interface between different media as
Sbc2; and (c) side of the conductor along the radius of the driven rotor as
Sbc3. The equations for these types of boundary conditions are as follows:
where
is the induced eddy current vector. In permanent magnet coupling, the main excitation source is the rotation of the permanent magnet, and the magnetic flux density vector
B at each position in the solution domain is expressed as follows:
The electromagnetic induction principle is used to further calculate the torque
dV (i.e., the differential element of volume) of the conductor tube, and the total torque is obtained by integration as follows:
where
rc1 and
rc2 are the outer and inner diameters of the conductor tube, respectively.
A simplified 3D model equivalent to the physical image of the permanent magnet coupling is shown in
Figure 9a; the structures with a negligible impact on the computational result are ignored [
29,
30,
31]. Considering the skin effect, tetrahedrons were used to divide the mesh, as shown in
Figure 9b. The material parameters of each part of the permanent magnet coupling were imported, and the boundary conditions were set for the solution. FEA of the CPMC under two working conditions with a slip of 50 rpm (rated) and 300 rpm (high load) was performed; the results are shown in
Figure 10,
Figure 11 and
Figure 12.
As shown in
Figure 10, the area with the highest magnetic flux density in the conductor barrel is the area where the permanent magnet is mapped to the conductor tube. The maximum values are 1.85, 1.78, and 1.75 T when the slip is 20, 50, and 300 rpm, respectively. When the slip increases, the intensity of the magnetic induction decreases. This is because as the slip increases, the magnetic field generated by the conductor tube also increases, which causes the working point of the permanent magnet (
Figure 6) to move toward the left; this, in turn, causes a reduction in the magnetic flux density.
As shown in
Figure 11, the direction of the eddy current in the conductor tube conforms to the direction determined by the right-hand spiral rule, and the center of the eddy current is located at the center of the two adjacent permanent magnet mapping areas. The magnetic field generated by the eddy current interacts with that of the permanent magnet to generate torque. The area with the highest eddy current density in the conductor tube is where the permanent magnet is mapped to the conductor tube. When the slip is 20, 50, and 300 rpm, the maximum eddy current density is
, and
, respectively. The induced current significantly increases with the increase in the slip, and this, in turn, increases the induced magnetic field. Therefore, the torque between the permanent magnet and the conductor tubes increases. However, the increase in the induced eddy current also increases the eddy current loss. According to
Figure 12, the maximum values of the eddy current loss density under the three slips are
, and
; an increase in the slip results in a significant increase in the eddy current loss. Hence, the heat generation of the permanent magnet coupling will significantly increase under high-load conditions.
The results of the FEA indicate that the eddy current that plays a decisive role in the torque transmission of the permanent magnet coupling is located in the area between the inner and outer permanent magnets (i.e., the mapping area of magnetic flux tube 1 on the conductor tube). The induced eddy current in the mapping areas of magnetic flux tubes 2 and 3 on the conductor tube is the smallest, two orders of magnitude lower than the maximum, and can be ignored in engineering calculations.
Figure 12 shows the test bench for the CPMC. A low-speed, high-torque motor is used as the driving element, which is connected to the CPMC via the input torque and speed sensor. The other side of the coupling is connected to the magnetic powder brake via the output torque and speed sensor. The engagement length between the conductor tube and the permanent magnet barrel can be adjusted using the screw structure. An infrared temperature sensor is placed outside the conductor tube to measure its temperature.
The results obtained using the computational model indicate that under the rated working condition of CPMC with a slip of 50 rpm, its torque is about 66 N.m, while the maximum torque is about 383 N.m, which is approximately 5.8 times the rated torque. As the input speed is the rated speed of the motor, and the relationship between the power of the PMC and the torque and input rotational speed
n is given by
, it can be deduced that the power consumed by the PMC is proportional to its torque; thus, the operating power of the PMC exceeds five times its rated power when it is close to overload. Therefore, the rated power of the drive motor of the test bench must exceed five times the rated power of the PMC to test its performance under extreme working conditions. In most previous studies, the rated power of the drive motor used in the test bench was close to the power of the PMC [
32,
33]. Such power can only test the characteristics of the PMC when the slip rate is within 0.3; it cannot test the ultimate performance of the coupling. The rated power of the prototype tested in this test bench was 5 kW. To test its characteristics under extreme working conditions, we chose a three-phase asynchronous motor with a rated power of 30 kW as the drive motor. The parameters are shown in
Table 3. The test bench could test the performance of the CPMC prototype in the full range of the slip rate from zero to one.
Speed torque sensors were arranged at both the input and output ends of the permanent magnet coupling to measure the differences between the input and output speeds and torques; that is, the differences in torque and speed are caused by mechanical friction and permanent magnet coupling, respectively. Thus, the error caused by mechanical friction can be eliminated.
Considering the working conditions with engagement lengths of 80, 60, 40, and 20 mm between the permanent magnet and conductor of the permanent magnet coupling, a comparison of the theoretical, simulation, and test data is presented in
Figure 13.
In
Figure 14, it can be seen that the transmission torque of the CPMC increases with the slip speed. When the slip speed approaches 800 rpm, the rate of increase in the transmission torque decreases; when the slip speed is approximately 800 rpm, the transmission torque reaches the maximum value; and when the slip speed continues to increase, the transmission torque decreases. This occurs because the increase in the slip speed causes an increase in the magnetic induction line cutting speed of the conductor tube as well as a gradual increase in the induced eddy current. Ampere’s force that is generated by the induced eddy current also increases gradually, leading to an increase in the transmission torque. However, when the slip speed is sufficiently high, the induced magnetic field formed by the induced eddy current will impose a reaction force on the original permanent magnet, which is reflected by the low-efficiency position of its working point. In addition, the skin effect gradually increases with the slip speed. Hence, the transmission torque gradually decreases when the slip speed is greater than 800 rpm.
Figure 14a–d indicate that for a constant slip speed and a decrease in the engagement length between the conductor tube and permanent magnet, the transmission torque of the permanent magnet coupling decreases. This is because the length of the magnetic field line cut by the conductor tube is shortened. In addition, the area where the conductor tube does not engage with the permanent magnet cannot generate eddy currents, leading to a decrease in the transmission torque. This characteristic of the CPMC can match the motor with different loads and limit the maximum transmission torque.
Furthermore, it is clear from
Figure 14 that the theoretical computational model, the finite element simulation results, and the test results are in good agreement. Nevertheless, the finite element simulation results are more accurate. When the slip speed is greater than 800 rpm, the permanent magnet completely slips away from the conductor disk, and the torque sensor of the test fails. Therefore, the load is stalled, and the conductor tube connected to the load stops rotating. The power on the motor side is completely converted into heat generated by the induced eddy current, and the coupling increases dramatically. When the temperature rises to a certain level, it causes irreversible damage to the permanent magnet because it prevents the motor from stalling.
We tested the CPMC from startup to stable operation under transmission powers of 2.5, 5, and 7.5 kW and recorded the temperature. The corresponding temperature–time curve obtained is shown in
Figure 15.
As shown in
Figure 15, an increase in the transmission power increases the heating of the CPMC. However, the temperature increase is relatively small for the 2.5 kW working condition; after 1100 s, the temperature increased from 20 to 33 °C to reach thermal equilibrium and then stopped increasing. For the 5 kW working condition, it took a relatively long time to reach the thermal equilibrium state; specifically, the temperature initially increased to 83 °C, and after 1800 s, it reached a stable state. While the temperature of the CPMC continued to rise under the 7.5 kW working condition, the heat and heat dissipation could not be balanced; hence, to prevent the temperature of the CPMC from exceeding the Curie temperature of the permanent magnet, which would result in overheating and subsequent failure of the permanent magnet, the test bench was closed when the temperature increased to 150 °C after running for 800 s.
3.2. Discussion and Comparison
Kano et al. [
5] proposed a PMC calculation model that uses the equivalent magnetic circuit approach. When the air gap of the PMC was 4 mm and the slip rate was 0.18, the error of their theoretical model was 9.7%, and when the air gap of the PMC was 8.5 mm and the slip rate was 0.15, the error was 11.3%. In short, the error of their calculation model increased with the air gap and slip rate. The theoretical model proposed by Cai and Wang [
11] had the largest error at the maximum point, with a value of about 13%. The theoretical model proposed by Mohammadi and Mirsalim [
34] had a higher accuracy under a low slip, but the error increased with the slip; the maximum error was 9% when the slip was 120 rpm. By contrast, the calculation model proposed in this article, when compared to the experimental values, has maximum errors of 2.3%, 2.5%, 3.6%, and 8.7% under engagement lengths of 80 mm, 60 mm, 40 mm, and 20 mm, respectively. The error increases with reductions in the engagement length, but compared with the above theoretical models, the theoretical model proposed in this article has higher accuracy.
The CPMC transmitted torque through the slip speed, and its torque density varied with the slip speed. The test results indicate that the torque densities of the CPMC were 9 Nm/kg when the slip speeds were 40 rpm.
Table 4 compares the torque density of an adjustable-speed permanent magnet eddy current coupling (AS-PMECC) at 40 rpm [
22], an aeronautical magnetic torque limiter at 20 °C [
35], a radial flux cylindrical permanent magnet coupling (RF-CPMC), and a flux-concentrating cylindrical permanent magnet coupling (FC-CPMC) based on a reference study [
36].
It can be seen from
Table 4 that the torque density of the AS-PMECC at 40 rpm is the lowest, and that of the CPMC at 40 rpm is slightly higher. However, compared to the magnetic torque limiter, RF-CPMC, and FC-CPMC, the torque density is much smaller for the CPMC and AS-PMECC.
In
Table 4, the CPMC and AS-PMECC are asynchronous PMCs, which provides them with the inherent advantages of speed adjustment, soft start, and overload protection. In contrast, the other three are synchronous PMCs, which provides better permanent magnet utilization than asynchronous PMC.