Torque Characteristics Analysis of Slotted-Type Axial-Flux Magnetic Coupler in the Misalignment State

Abstract

In this article, a simple and practical magnetic equivalent charge model is proposed to predict the torque of a slotted-type axial-flux magnetic coupler (SAMC) under conditions of radial misalignment, angle misalignment, and synthetic misalignment. The magnetic field generated by the permanent magnet (PM) disk and the induced magnetic field generated by the slotted conductor sheet (CS) are equivalent to the surface magnetic charge, respectively. Particularly, the induced magnetic field produced by eddy current considering skin depth in the conductor sheet is introduced into the magnetic equivalent charge model. Combined with Coulomb’s law of magnetic field, the formulas of torque and axial force are both derived. Using this method, the torques in three cases of misalignment are calculated. Finally, the effectiveness of the model is verified by the finite element method (FEM) and experiment; the results calculated by the magnetic equivalent charge model are basically consistent with those from the finite element method and experiment. The derived formula is suitable for small air gaps, small slip rates, and small radial deflection distances. Additionally, the limitations of the method proposed are discussed, which is of great help for understanding the torque transmission of the magnetic coupler in the misalignment state.

1. Introduction

A magnetic coupler is a non-contact transmission device driven by magnetic force. In addition, the torque and speed can also be adjusted by adjusting the air gap length of the disk-magnetic coupler or overlap area of the cylindrical-magnetic coupler. Because magnetic couplers are superior to traditional mechanical transmission devices in terms of energy savings, environmental protection, corrosion resistance, ease-of-maintenance, and overload protection, they have been widely used in many industrial applications [1,2,3,4,5,6].

Currently, research on the magnetic coupler mainly focuses on torque calculation, temperature rise calculation, and structural optimization design under conditions of complete alignment [7,8,9,10,11]. But, in actual working processes, due to manufacturing, installation, thermal deformation, and other factors, it is inevitable that there will be misalignments between the two rotors [12,13,14,15,16]; that is to say, usually there are position angle and displacement differences between the axes of two of rotors, which affect the torque greatly. However, so far there have been few studies on the torque when the magnetic coupler is in misalignment. In fact, for the magnetic coupler, it has been reported that a slotted CS topology can significantly increase torque density and reduce temperature rise compared with a non-slotted topology [17]. Therefore, considering the complexity of the slotted CS structure in slotted-type axial-flux magnetic couplers and the different positions between the PM and slotted CS leads to a variation in the air gap magnetic field in the misaligned state. Therefore, the torque calculation methods used for misalignment must be different to methods used for those without misalignment.

Usually, there are two primary theoretical methods for torque computation. The first is the magnetic vector method, which utilizes the analytical model of the magnetic vector derived from Maxwell’s equations and appropriate boundary conditions. In [18,19], the magnetic field distribution and torque of the magnetic coupler were obtained by the magnetic vector method, to simplify the calculation, a magnetic vector layer model was established in the cartesian coordinate system, when the magnetic coupler is expanded along the mean radius. The other method is the more widely used equivalent magnetic circuit method, which establishes the analysis model by analyzing the two-dimensional magnetic flux path direction. The correlation equation of magnetic flux by combining with Kirchhoff’s law and Ampere’s law is then obtained. In [20,21], the torque and magnetic field intensity of the magnetic coupler, which the induced current introduced, were calculated by the equivalent magnetic circuit method. Also, in [22,23], the torque and magnetic field intensity were both calculated by the equivalent magnetic circuit method for the magnetic rotating magnetic couplers, especially when the conductor disk was not parallel to the permanent magnet disk so that there was a rotating angle between the two rotors. However, the three methods mentioned above can only be applied in some cases where the axis of the permanent magnet rotor coincides with that of the conductor rotor. In this case, the magnetic couplers can be expanded and linearized along the mean radius. But when there is an offset in radius direction of the rotor, that is to say, the axis of the PM rotor does not coincide with that of the slotted CS rotor, the theoretical methods mentioned above have not been applied directly. It is necessary to study the torque characteristics of the slotted-type axial-flux magnetic coupler in the misalignment state, which often occurs in practical engineering applications.

In order to analyze the transmission characteristics of the SAMCs when in misalignment, a new equivalent magnetic charge method that introduces the induced magnetic field is proposed. According to the theory of magnetic equivalent charge, the distribution model of the positive and negative magnetic charges on two respective surfaces of the PMs and slotted CS can be described based on magnetization direction of PMs and reaction magnetic field direction generated by an eddy current on the slotted CS. The magnetic charge distribution of SAMC can be obtained accordingly when there is a radial misalignment, an angle misalignment, or a synthetic misalignment of SAMCs. After the magnetic induction intensity is calculated by the equivalent magnetic circuit method and considering the skin effect, the magnetic force between two surface magnetic charges can be calculated using Coulomb’s law for each surface separately. From the above, the torque expression of SAMC in a misaligned state can be derived, and the torque and axial forces are calculated by the new method proposed. The results obtained by the method proposed are compared with those simulated by FEM and an experiment to verify the correctness.

2. Theory

Figure 1 is the 3D-model of the SAMC, where axially magnetized PMs are alternately placed on the PM disk yoke iron, and the slotted CS is embedded in the slotted CS yoke iron with iron teeth. Figure 2 illustrates the structure and geometrical parameters of the slotted CS and the PM disk. In order to achieve better magnetic transmission efficiency and system performance, the selected SAMC physical parameters are shown in

Table 1. Specifications of the studied coupler.

Parameter	Value
Number of PM pole pairs, p	9
Thickness of PMs, hm/mm	10
Thickness of PM disk yoke iron, hmb/mm	10
Outer radius of slots, Rb2/mm	10
Thickness of the conductor, hc/mm	10
Angle of pole-arc, θm	20°
Inner radius of the PMs, Rm1/mm	85
Outer radius of the PMs, Rm2/mm	120
Inner radius of conductor, Rc1/mm	75
Outer radius of conductor, Rc2/mm	130
Inner radius of slots, Rb1/mm	80
Outer radius of slots, Rb2/mm	125
Angle of a fan-shaped conductor, θc	15°
Angle of a fan-shaped solt, θs	7.5°
Permeability of vacuum, μ0/(H/m)	4π × 10−7
Coercivity of the PMs, Hc/(A/m)	−8.68 × 105
Conductivity of the conductor, σc/(S/m)	5.7 × 107

.

2.1. The Influence of the Misalignment of SAMC

In order to change the output speed and torque, the length of the air gap between the PM disk and the slotted CS is adjusted, which introduces axial misalignment is neglected here. The misalignment of SAMC can be categorized into three types (shown in Figure 3): radial misalignment δ, angle misalignment α, and synthetic misalignment δ, α. Because the torque is dependent on the magnetic field intensity, when the influence of three misalignment forms on the output torque is discussed, the possible misalignment of the magnetic flux paths of the SAMC needs to be analyzed. Figure 4 shows the magnetic flux paths of one pole pair during complete alignment, wherein, the magnetic flux path I is the main flux path (shown as a solid line) and leakage magnetic flux path (shown as a dotted line) is distributed vertically through the copper conductor along the axis, respectively. In addition, the magnetic flux paths I and II also show the distributions of two different positions of the main magnetic flux path and leakage magnetic flux path along the radius direction. When there is the misalignment of the SAMC, both the main magnetic flux and the leakage magnetic flux may increase or decrease according to the three misalignment cases as follows:

(1)

Radial misalignment, as shown in Figure 3a: when the axis of the PM disk is parallel to that of the slotted CS, and there is a deviation distance between the two axes, which causes the decrease of the positive area of slotted CS disk, the main magnetic flux decreases but the leakage magnetic flux increases, so the magnetic-field-line-cutting range is decreased, and the torque of the SAMC is also decreased.

(2)

Angle misalignment, as shown in Figure 3b: there is a certain inclination angle between the two axes of the PM disk and the slotted CS in a radial direction. Due to the inclination angle, some parts of the PM disk are close to the slotted CS, while other parts are away from the slotted CS. The smaller the air gap length, the more the main flux, and the more magnetic field lines the slotted CS crosses, so the larger the output torque and vice versa. When the air gap is smaller, the torque rises quickly due to the reduction of the leakage magnetic flux in all directions. Therefore, when there is an angle misalignment, the side where the air gap length decreases has a greater effect on the output torque than the side where the air gap length increases. That is, the output torque increases with the increase of the angle.

(3)

Synthetic misalignment, as shown in Figure 3c: there is not only a certain inclination angle but also a deviation distance between two axes, The magnitude of torque should be affected by both the angle deflection and the radial deflection. In this case, the relationship is more complex.

2.2. Basic Assumptions and Model Building

To simplify the calculation, the following assumptions are made:

(1)

All magnets on the PM disk are uniformly magnetized along the axis direction.

(2)

The magnetic charge on the surfaces of the PMs is evenly distributed, and the magnetic charge is only distributed on the surfaces perpendicular to the direction of magnetization.

(3)

Since the magnetic field generated by the induced current in the slotted CS is constantly under a certain stable speed difference, the slotted CS that generates an inducedmagnetic field is equivalent to the PM disk when in movement.

(4)

The yoke iron of the PM and slotted CS disks is too thick to cause magnetic saturation; meanwhile, the effect of yoke iron magnetization on the electromagnetic moment is ignored.

Figure 5a is the 3D model of SAMC when synthetic misalignment. In order to analyze the transmission characteristics of the SAMC, a three-dimensional cartesian coordinate system is set on the bottom surface of the PMs, and the center O of the bottom surface is set as the origin of the system. Let the upper surface of the slotted CS be 1, the bottom surface 2, the upper surface of the PM disk 3, and the bottom surface 4. As shown in Figure 5b, surface 2 is inclined to surface 3 at an angle α, and the center O’ of surface 2 is offset to the z-axis at a given distance δ. In order to facilitate the observation of each set of parameters, surface 2 and surface 3 are taken out separately, and a model diagram is established, shown in Figure 5b.

Figure 6 is the magnetic charge distribution and eddy current distribution of one pair of poles when the relative position angle of the PM disk and the slotted CS φ is 0. According to the theory of the magnetic equivalent charge, the arrow of the magnetization direction of the PM is equivalent to the positive magnetic charge, and the other end of magnetization direction is equivalent to the negative magnetic charge. Therefore, the magnetic charge on the sides of the N pole is opposite to that on the sides of the S pole. In order to express the distribution of magnetic charge on the slotted CS in the SAMC, the induced magnetic field generated by the copper conductor is equivalent to an induction magnet, and the north and south magnetic poles of the induction magnet correspond to positive and negative magnetic charges, respectively (see Figure 6). Therefore, the force between the PM disk and the slotted CS is produced because of the interaction between the magnetic charge distributed on surfaces 1 and 2 of the equivalent PMs and the magnetic charge distributed on the surfaces 3 and 4 of the PMs. The opposite magnetic charge toward each other between the PMs and CS produces the attraction, while the same magnetic charge toward each other produces the repulsive force.

According to Coulomb’s law, the force between magnetic charge is

$$d\overrightarrow{F_{ab}}={\displaystyle \frac{d{q}_{a}d{q}_b}{4\pi {\mu }_0\overrightarrow{d_{ab}^2}}}$$

(1)

where $q_a$ and $q_b$ are the magnetic charge at point a and b, respectively, $\overrightarrow{d_{ab}}$ is the position vector from point a to point b.

In order to solve the force between the PM disk and the slotted CS, take any point a on surface 2 and any point b on surface 3, according to the definition formula of magnetic charge surface density, q_a and q_b of point a and point b are represented, respectively.

$$\left\{\begin{array}{l}d{q}_a={\sigma }_2\left|\overrightarrow{r_2}\right|d{\beta }_{2}d\left|\overrightarrow{r_2}\right|\\ d{q}_b={\sigma }_3\left|\overrightarrow{r_3}\right|d{\beta }_{3}d\left|\overrightarrow{r_3}\right|\end{array}\right.$$

(2)

where σ₂ and σ₃ are the magnetic charge surface densities of surfaces 2 and 3, $\overrightarrow{r_2}$ is the vector quantity from the z axis to the point a, β₂ is the positive angle of $\overrightarrow{r_2}$ with respect to the x-axis, see Figure 7. $\overrightarrow{r_3}$ is the vector quantity from the z axis to the point b, and β₃ is the angle between the vector $\overrightarrow{r_3}$ and the positive of the x-axis.

Figure 7 shows the geometric relationship after the projection of the slotted CS on the x–y plane when there is synthetic misalignment; according to the trigonometric function relationship the distance r₂ is expressed as

$${\left|\overrightarrow{r_2}\right|}^2={\delta }^2+{l_1}^2+2\delta l_1\cos \left[{\displaystyle \frac{\arctan (\cos \alpha )}{{\beta }_c}}\right]$$

(3)

wherein,

$$l_1=\left|\overrightarrow{R_{\mathrm{ca}}}\right|\times \sqrt{{\cos }^2({\beta }_c){\cos }^2(\alpha )+{\sin }^2({\beta }_c)}$$

(4)

where $\overrightarrow{R_{\mathrm{c}a}}$ is the vector quantity from the center O’ of the surface 2 to point a, β_c is the angle between $\overrightarrow{R_{\mathrm{c}a}}$ and the plane of the y and z axes.

Thus, the angle β₂ between $\left|\overrightarrow{r_2}\right|$ and the x-axis is expressed as follows:

$${\beta }_2=\arccos {\displaystyle \frac{l_1\sin \left[\frac{\arctan (\cos \alpha )}{{\beta }_c}\right]}{\sqrt{{\delta }^2+{l_1}^2+2\delta l_1\cos \left[\frac{\arctan (\cos \alpha )}{{\beta }_c}\right]}}}$$

(5)

The length of the air gap between the two disks varies circumferentially, and the corresponding air gap length at any position is

$$h_g=h+\left|\overrightarrow{R_{\mathrm{ca}}}\right|\sin \alpha \cos {\beta }_c$$

(6)

where h_g is the length of the air gap from point a and surface 3, h is the distance from O’ to surface 3.

According to the Ampere’s loop theorem of magnetic charge density, the relationship between the magnetic charge surface density and the magnetic induction intensity is as follows:

$$\left\{\begin{array}{c}{\sigma }_2=B_c\\ {\sigma }_3=B_r\end{array}\right.$$

(7)

where B_c is the magnetic induction intensity generated by the conductor cutting the magnetic field line under a constant slip rate s, and B_r is the residual magnetic induction intensity of the PM material.

The magnetic induction intensity B_c can be calculated by the equivalent magnetic circuit method, using a combination of Kirchhoff’s law and Ampere’s law, the formula [17] is as follows:

$$B_c=\left\{\begin{array}{ccc}{\displaystyle \frac{{\mu }_0{J}_{\mathrm{av}}{h}_c}{2\pi }}\ln {\displaystyle \frac{({\tau }_m/2-x)({\tau }_p-{\tau }_{\mathrm{s}2}-x)}{({\tau }_{s1}-x)({\tau }_p-{\tau }_m/2-x)}}& when& 0\le x\le {\tau }_{s1}\\ {\displaystyle \frac{{\mu }_0{J}_{\mathrm{av}}{h}_c}{2\pi }}\ln {\displaystyle \frac{{\tau }_p-{\tau }_{\mathrm{s}2}-x}{{\tau }_p-{\tau }_m/2-x}}& when& {\tau }_{s1}\le x\le {\tau }_m/2\\ {\displaystyle \frac{{\mu }_0{J}_{\mathrm{av}}{h}_c}{2\pi }}\ln {\displaystyle \frac{(x-{\tau }_{\mathrm{s}1})({\tau }_p-{\tau }_{\mathrm{s}2}-x)}{(x-{\tau }_m/2)({\tau }_p-{\tau }_m/2-x)}}& when& {\tau }_m/2\le x\le {\tau }_p-{\tau }_m/2\\ {\displaystyle \frac{{\mu }_0{J}_{\mathrm{av}}{h}_c}{2\pi }}\ln {\displaystyle \frac{x-{\tau }_{\mathrm{s}1}}{x-{\tau }_m/2}}& when& {\tau }_p-{\tau }_m/2\le x\le {\tau }_p-{\tau }_{s2}\\ {\displaystyle \frac{{\mu }_0{J}_{\mathrm{av}}{h}_c}{2\pi }}\ln {\displaystyle \frac{(x-{\tau }_{\mathrm{s}1})(x-{\tau }_p+{\tau }_m/2)}{(x-{\tau }_m/2)(x-{\tau }_p+{\tau }_{\mathrm{s}2})}}& when& {\tau }_p-{\tau }_{s2}\le x\le {\tau }_p\end{array}\right.$$

(8)

where J_av is the average vortex density; τ_p is the pole moment of PM, it is calculated as ${\tau }_p={\displaystyle \frac{\pi (R_{m1}+R_{m2})}{2p}}$; τ_m is the average polar arc length of PM in SAMC mentioned above, ${\tau }_m={\tau }_p$; τ_s1 and τ_s2 are, respectively the average polar arc lengths of the iron teeth corresponding to the N-pole PM and S-pole PM.

Substitute Equations (2)–(8) into Equation (1), the differential form of the magnetic force between the magnetic charge at point a on surface 2 and point b and on surface 3 is

$$d\overrightarrow{F_{ab}}={\displaystyle \frac{1}{4\pi {\mu }_0}}{\displaystyle \frac{{\sigma }_2\left|\overrightarrow{r_2}\right|d{\beta }_{2}d\left|\overrightarrow{r_2}\right|{\sigma }_3\left|\overrightarrow{r_3}\right|d{\beta }_{3}d\left|\overrightarrow{r_3}\right|}{{\left|\overrightarrow{d_{23}}\right|}^3}}\overrightarrow{d_{23}}$$

(9)

wherein,

$${\left|\overrightarrow{d_{23}}\right|}^3=\left[{\left(\left|\overrightarrow{r_3}\right|\cos {\beta }_3-\left|\overrightarrow{r_2}\right|\cos {\beta }_2\right)}^2+{h_g}^2\right.+{\left.{\left(\left|\overrightarrow{r_3}\right|\sin {\beta }_3-\left|\overrightarrow{r_2}\right|\sin {\beta }_2\right)}^2\right]}^{\frac{3}{2}}$$

(10)

Then, the magnetic force can be decomposed into the components as

$$d{F}_{abz}=d\overrightarrow{F_{ab}}·\overrightarrow{j}={\displaystyle \frac{B_c{B}_r}{4\pi {\mu }_0}}{\displaystyle \frac{\left|\overrightarrow{r_2}\right|d{\beta }_2\left|\overrightarrow{r_3}\right|d{\beta }_{3}d\left|\overrightarrow{r_2}\right|d\left|\overrightarrow{r_3}\right|}{{\left|\overrightarrow{d_{23}}\right|}^3}}{h}_g$$

(11)

$$d\overrightarrow{F_{ab\theta }}={\displaystyle \frac{B_c{B}_r}{4\pi {\mu }_0}}{\displaystyle \frac{\left|\overrightarrow{r_2}\right|d{\beta }_2\left|\overrightarrow{r_3}\right|d{\beta }_{3}d\left|\overrightarrow{r_2}\right|d\left|\overrightarrow{r_3}\right|}{{\left|\overrightarrow{d_{23}}\right|}^3}}\left(\overrightarrow{r_3}-\overrightarrow{r_2}\right)$$

(12)

where dF_abz is the axial component on the z axis, $d\overrightarrow{F_{abx}}$ is the circumferential component, $\overrightarrow{j}$ is the unit vector in the z direction.

From Equation (12), the torque between point a and point b can be obtained. But the torque generated by N-pole or S-pole should be solved separately, and then be summed, and then the torque between surfaces 2 and 3 is

$$T_{23}=2p{\displaystyle \sum _{i=1}^{2p}{\left(-1\right)}^{i-1}}{\displaystyle {\int }_{R_{c1}}^{R_{c2}}d\left|\overrightarrow{r_2}\right|{\displaystyle {\int }_{R_{m1}}^{R_{m2}}d\left|\overrightarrow{r_3}\right|{\displaystyle {\int }_{\phi }^{\lambda }d{\beta }_3{\displaystyle {\int }_{{\phi }_{i-1}}^{{\phi }_i}\frac{B_c{\left|\overrightarrow{r_2}\right|}^2{B}_r{\left|\overrightarrow{r_3}\right|}^2\sin \left({\beta }_3-{\beta }_2\right)}{4\pi {\mu }_0{\left|\overrightarrow{d_{23}}\right|}^3}d{\beta }_2}}}}$$

(13)

wherein,

$$\lambda ={\displaystyle \frac{2\pi }{m}}+\phi$$

(14)

$$\left\{\begin{array}{l}{\phi }_i={\displaystyle \frac{{\theta }_{c}m}{{\theta }_c+{\theta }_s}}\times {\displaystyle \frac{2\pi \left(2p-i+1\right)}{m}}\\ {\phi }_{i-1}={\displaystyle \frac{{\theta }_{c}m}{{\theta }_c+{\theta }_s}}\times {\displaystyle \frac{2\pi (2p-i)}{m}}\end{array}\right.i=1,2,\mathrm{3...2}p)$$

(15)

where m is the number of slots in the slotted CS, θ_c is the pole arc angle of the conductor, θ_s is the angle of a fan-shaped slot (see Figure 2b).

Suppose $\left|\overrightarrow{d_{24}}\right|$, $\left|\overrightarrow{d_{13}}\right|$, and $\left|\overrightarrow{d_{14}}\right|$ are the position vectors between surfaces 2 and 4, between surfaces 1 and 3, and between surfaces 1 and 4, they can be described as

$${\left|\overrightarrow{d_{24}}\right|}^3=\left[{\left(\left|\overrightarrow{r_3}\right|\cos {\beta }_3-\left|\overrightarrow{r_2}\right|\cos {\beta }_2\right)}^2+{\left(h_g+h_m\right)}^2\right.+{\left.{\left(\left|\overrightarrow{r_3}\right|\sin {\beta }_3-\left|\overrightarrow{r_2}\right|\sin {\beta }_2\right)}^2\right]}^{\frac{3}{2}}$$

(16)

$${\left|\overrightarrow{d_{13}}\right|}^3=\left[{\left(\left|\overrightarrow{r_3}\right|\cos {\beta }_3-\left|\overrightarrow{r_2}\right|\cos {\beta }_2\right)}^2+{\left(h_g+h_{c\mathrm{e}}\cos \alpha \right)}^2\right.+{\left.{\left(\left|\overrightarrow{r_3}\right|\sin {\beta }_3-\left|\overrightarrow{r_2}\right|\sin {\beta }_2\right)}^2\right]}^{\frac{3}{2}}$$

(17)

$${\left|\overrightarrow{d_{14}}\right|}^3=\left[{\left(\left|\overrightarrow{r_3}\right|\cos {\beta }_3-\left|\overrightarrow{r_2}\right|\cos {\beta }_2\right)}^2+{\left(h_g+h_{ce}\cos \alpha +h_m\right)}^2\right.+{\left.{\left(\left|\overrightarrow{r_3}\right|\sin {\beta }_3-\left|\overrightarrow{r_2}\right|\sin {\beta }_2\right)}^2\right]}^{\frac{3}{2}}$$

(18)

where h_ce is the effective thickness of the slotted CS.

With the increase of the rotational speed difference and the number of poles, the frequency of eddy current on the slotted CS increases, and the skin effect is generated, so that most of the eddy current is concentrated on the surface of the slotted CS. The skin depth can be given by [24].

$$\tau =\sqrt{{\displaystyle \frac{15}{\pi p(n_1-n_2){\sigma }_c{\mu }_0}}}$$

(19)

$$\Delta =\tau (1-{\displaystyle \frac{1}{e}})$$

(20)

$$h_{ce}=min\left\{h_c,\Delta \right\}$$

(21)

where n₁ is the driving speed of the disk, n₂ is the driven speed of the disk., and σ_c is the conductor conductivity, τ is the skin depth,

Δ is the effective depth of the vortex density, h_ce is the effective thickness of the slotted CS.

Similar to Equation (13), T₂₄, T₁₃, T₁₄ are the torques between surfaces 2 and 4, between surfaces 1 and 3, and between surfaces 1 and 4, respectively, and can be described as

$$T_{24}=2p{\displaystyle \sum _{i=1}^{2p}{\left(-1\right)}^{i-1}}{\displaystyle {\int }_{R_{c1}}^{R_{c2}}d\left|\overrightarrow{r_2}\right|{\displaystyle {\int }_{R_{m1}}^{R_{m2}}d\left|\overrightarrow{r_3}\right|{\displaystyle {\int }_{\phi }^{\lambda }d{\beta }_3{\displaystyle {\int }_{{\phi }_{i-1}}^{{\phi }_i}\frac{B_c{\left|\overrightarrow{r_2}\right|}^2{B}_r{\left|\overrightarrow{r_3}\right|}^2\sin \left({\beta }_3-{\beta }_2\right)}{4\pi {\mu }_0{\left|\overrightarrow{d_{24}}\right|}^3}d{\beta }_2}}}}$$

(22)

$$T_{13}=2p{\displaystyle \sum _{i=1}^{2p}{\left(-1\right)}^{i-1}}{\displaystyle {\int }_{R_{c1}}^{R_{c2}}d\left|\overrightarrow{r_2}\right|{\displaystyle {\int }_{R_{m1}}^{R_{m2}}d\left|\overrightarrow{r_3}\right|{\displaystyle {\int }_{\phi }^{\lambda }d{\beta }_3{\displaystyle {\int }_{{\phi }_{i-1}}^{{\phi }_i}\frac{B_c{\left|\overrightarrow{r_2}\right|}^2{B}_r{\left|\overrightarrow{r_3}\right|}^2\sin \left({\beta }_3-{\beta }_2\right)}{4\pi {\mu }_0{\left|\overrightarrow{d_{13}}\right|}^3}d{\beta }_2}}}}$$

(23)

$$T_{14}=2p{\displaystyle \sum _{i=1}^{2p}{\left(-1\right)}^{i-1}}{\displaystyle {\int }_{R_{c1}}^{R_{c2}}d\left|\overrightarrow{r_2}\right|{\displaystyle {\int }_{R_{m1}}^{R_{m2}}d\left|\overrightarrow{r_3}\right|{\displaystyle {\int }_{\phi }^{\lambda }d{\beta }_3{\displaystyle {\int }_{{\phi }_{i-1}}^{{\phi }_i}\frac{B_c{\left|\overrightarrow{r_2}\right|}^2{B}_r{\left|\overrightarrow{r_3}\right|}^2\sin \left({\beta }_3-{\beta }_2\right)}{4\pi {\mu }_0{\left|\overrightarrow{d_{14}}\right|}^3}d{\beta }_2}}}}$$

(24)

Then, the electromagnetic torque T_e transmitted is

$$T_e=T_{23}+T_{24}+T_{13}+T_{14}$$

(25)

The torque calculation method under synthetic misalignment is described above. If the torque in radial misalignment or angle misalignment should be considered separately, only take δ = 0 or α = 0 in the formula.

3. Model Verification and Discussion

3.1. Establishment of FEM Model

The basic structure and physical parameters of the SAMC studied in this section are the same as those described in Table 1 and Figure 1, which mainly include PM disk yoke iron, PM, slotted CS, and slotted CS yoke iron. The prototype is made using fan-shaped NdFeB magnets, pure copper plates, and industrial pure iron (DT4) plates. Figure 8 shows the FEM model of SAMC. Simulation software is used to generate a sufficiently thin grid by using a fixed computational grid method, so that the conductor layer can maintain numerical stability.

3.2. Analysis of FEM Model

Figure 9 shows the direction distribution of the magnetic flux path inside the SAMC in the case of complete alignment. As predicted in Figure 4, the flux path starts from the N pole of the PM disk, passes through the air gap to the slotted copper CS and the slotted CS disk yoke iron, then passes through the air gap again to the S-pole of the PM disk, and finally returns to the N-pole after passing through the PM disk yoke iron. And, because the magnetic permeability of the iron tooth is greater than that of the copper conductor, it leads to a change in the direction of the magnetic field lines, so that the magnetic field line tends to pass through the iron tooth, thereby the torque density of the SAMC increases and the temperature rise decreases in the working process.

In addition, the leakage magnetic flux path is disorganized, but the more the main flux path, the less the leakage magnetic flux path, so the magnetic field line density through the slotted copper CS can be used to verify the misalignment of the magnetic flux path.

Figure 10 shows the distribution of magnetic field lines inside the SAMC in the case of radial misalignment, and it can be seen that the PM disk deviates from the distant part of the slotted copper CS, compared with Figure 9, the density of magnetic field line of force inside the slotted copper CS, which is not directly aligned with PM disk decreases significantly. That is, the leakage magnetic flux path in this part increased, which was consistent with the prediction.

Figure 11 shows the distribution of magnetic field lines in the case of angle misalignment, and it can be seen that in the distant part where the distance between the PM disk and the slotted copper CS is far, the magnetic field line density decreases compared with Figure 9. That is, the leakage magnetic flux path in this part increased, which was consistent with the prediction. However, for the adjacent part close to the PM disk and the slotted copper CS, the magnetic field line density increases significantly compared with Figure 9. That is, the leakage magnetic flux path in this part increased, which was consistent with the prediction.

Figure 12 shows the trend of the eddy current in the CS. It can be seen that there are different eddy current paths in the CS, mainly the elliptical eddy current path between two tooth slots and the elliptical eddy current path between the two fan-shaped copper conductors. They correspond to the three paths, a, b, and c, shown in Figure 6. These distributions are caused by changes in the area of the conductor and the iron teeth facing PM during motion.

3.3. Establishment of the Test Platform

Figure 13 shows the experimental measurement platform, including a prime motor, two torque sensors, and a load motor, which is used to provide load. The prime motor is connected to the torque sensor, and then the SAMC is installed on the active end, and both the PM disk and the slotted CS disk can be used as the active end. In this experiment, the prime motor and the PM disk are used as the active end, and the slotted CS disk is connected to the torque sensor and then connected to the load to form the driven end. The input speed is 1200 rpm, and the air gap length and the output speed are controlled to obtain different output torque ratings. At the same time, a set of experiments is repeated several times to ensure the accuracy of the data. In addition, the performance of the PMs is affected by the high temperature, if the eddy current in the conductor disk is too large; as a result, the test is performed at low slip speeds.

3.4. Comparison and Discussion with 3-D FEM Method and Experimental Results

In order to verify the applicability of the model proposed in this paper, the torque sizes of different radial and angle misalignments under different air gaps calculated by the theoretical method are compared with those simulated by the FEM. The results are shown in Figure 14 and Figure 15.

Figure 14 shows torque curves when in radial misalignment. The average torque is almost constant when the radial misalignment value δ is small, while the average torque begins to decrease when the radial misalignment value δ is large. This is because in the design of the SAMC, the radius of the slotted copper CS is deliberately enlarged compared with the radius of the PM disk in order to deflect the current on slotted copper CS and form eddy currents. Therefore, when the radial misalignment value δ is relatively small, most eddy currents can still be formed, and the torque is almost unchanged. However, when the radial misalignment value δ is relatively large, the torque begins to decrease.

Figure 15 shows the torque curves when the angle is misaligned; the torque will increase with the increasing of the angle misalignment value α. The influence of angle misalignment on torque is greater than that of radial misalignment in the cases of small range misalignment and small radial misalignment. In addition, in the case of 4 mm in Figure 15, when the angle misalignment is large, a collision will occur between the PM rotor and the CS rotor. Therefore, when the air gap length is 4 mm, there is no data for 2 mm and 2.5 mm.

The torque variation trend of the two misalignments can be well presented by theoretical calculation. They have good consistency when there is a 4 mm air gap length, but the torque generated by the slotted CS yoke iron is not considered in this paper, resulting in a small theoretical value. In the cases of 6 mm and 8 mm, the results obtained by the theoretical model is larger than the results obtained by simulation because it is assumed that the magnetic flux passes perpendicular to the air gap and the slotted copper CS when calculating the torque, but when the air gap is larger, there is more leakage magnetic flux, so in the case of a large air gap, the calculated induced magnetic field value is larger, resulting in a larger torque value calculated theoretically. And the larger the air gap, the more it is inaccurate.

Figure 16 is the torque curves when synthetic misalignment of the inclination angle is α = 0.5° and slip rate is s = 0.125. The overall trend is the same as in Figure 15, but the torque value at each point is larger than the corresponding torque value in Figure 15. The results by simulation and calculation are still in good agreement when the air gap is 4 mm, but the error when there is a 6 mm air gap is larger than that in case of radial misalignment or angle misalignment in Figure 14 and Figure 15, which is due to the superposition of two errors in radial misalignment and angle misalignment.

Figure 17 shows the relationship between torque and air gap length obtained by theoretical calculation when α = 0°, δ = 0 mm, compared with FEM results and experimental results. It can be seen that when the length of the air gap increases, the deviation of the theoretical calculation results will be larger; when h = 6 mm, the error is 10.98%. This is still because the main magnetic flux will gradually bend and decrease under a larger air gap thickness, and the magnetic leakage will increase, resulting in a larger induced magnetic flux calculation and thus a larger torque value.

Figure 18 shows the relation between torque and air gap length when α = 1°, δ = 12 mm. It can be seen that the torque obtained by theoretical calculation is still greater than that obtained by FEM, and the error becomes larger; when h = 6 mm, the error is 13.78%. This is because the induced magnetic field will change after the misalignment occurs, which leads to the inaccuracy of the theoretically calculated induced magnetic field.

The slip rate is the important index for SAMC, and the slip rate has a significant impact on both eddy current density and torque characteristics, so the influence of the slip rate on the torque is studied. Figure 19 shows the relation between torque and slip rate under the α = 0°, δ = 0 mm, when air gap length h = 4 mm. From Figure 19, the torque obtained by the theoretical calculation, simulation, and experimental have good consistency under low slip rate; when the slip rate is 0.104, the error is 11.66%.

Figure 20 shows the relation between torque and slip rate under the synthetic misalignment of α = 0.5°, δ = 4 mm, when h = 4 mm. From Figure 20, torque figures obtained by the theoretical calculation and simulation also have good consistency under low slip rate; when the slip rate is 0.0625, the error is 18.9%. But because the SAMC usually works at low slip rate to reduce eddy current loss, this method proposed can also be used to calculate torque in misalignment.

4. Conclusions

In order to analyze the torque characteristic of the SAMC in a state of misalignment, a new magnetic equivalent charge model is presented. And, the induced magnetic field produced by eddy current, considering skin depth in the slotted CS is introduced into the magnetic equivalent charge model proposed, the torque and axial force formulas under conditions of radial misalignment, angle misalignment, and synthetic misalignment are derived.

In order to verify the effectiveness of the model, the magnetic fields under three kinds of misalignments and the distribution of eddy current in the slotted CS and the torque transmitted by the SAMC are simulated by FEM, which are in good agreement with the theoretical analysis.

The test results show that radial misalignment reduces torque, while angle misalignment increases it. The influence of angle misalignment on torque is greater than that of radial misalignment in the cases of small range misalignment and small radial misalignment. Torque decreases with the increase of air gap length and increases with the increase of slip rate. The results are used to analyze the torque transmission of various misalignment situations that often occur in the practical application of SMC and have certain guiding significance in situations where the torque transmitted in practice does not meet the appropriate requirements.

For radial, angle, and synthetic misalignments at an air gap length of 4 mm, this method has good accuracy. However, for larger air gap lengths, the theoretical results are too large. In such cases, theory can be applied to large air gap states by using semi-analytical methods to obtain induce magnetic field magnitude, obtained from FEM. Furthermore, this method still yields satisfactory results over a wide range of slip rate velocities, compared with existing FEM methods.