Analysis of Two-dimensional Cogging Torque of Permanent Magnet Spherical Motor with Soft Magnetic Composite Material Core

Abstract

The use of soft magnetic composite materials (SMCM) core in the permanent magnet spherical motor (PMS_pM) can increase the air gap magnetic flux density and torque density. However, it will also cause the distortion of the magnetic flux density, leading to the generation of cogging torque. In this paper, an analytical model of cogging torque for PMS_pM with iron core is proposed. Firstly, based on the spatial distribution of the air gap, the distribution function characterizing the tooth slot structure in both the longitude and latitude direction is obtained through Fourier expansion. Secondly, an analytical model of air gap magnetic flux density is obtained based on the distribution function of the tooth slot structure. Next, an analytical model for the rotation and yaw cogging torque is established based on the virtual work method. Finally, the analytical results are verified by finite element analysis and experimental results. The model can quickly obtain the relationship between the cogging torque and structural parameters such as permanent magnet thickness, air gap length, and stator slot opening angle, which lays a certain theoretical foundation for the structural optimization design of the PMS_pM with SMCM cores.

Keywords

soft magnetic composite material cogging torque virtual work method

Introduction

With the advancement of modern industry, an increasing number of scenarios in aerospace, military, medical, and other fields require the accomplishment of multi-degree-of-freedom operation (DOF). However, current multi-DOF operation relies on multiple motors in conjunction with complex mechanical transmission structure, each of which performs only single-DOF operation.¹ Deformation of mechanical coupling structure in extreme environments is a common failure of systems operating in multi-DOF.² A single axis spherical motor³ that can achieve three-DOF movement is expected to reduce the mechanical components of the system, thereby reducing its failure rate and improving its reliability. However, the unique three-dimensional structure of spherical motor increases it complexity in terms of body structure design, electromagnetic analysis, material processing technology, etc. The spherical motor is still in laboratory research and exploratory stage.

Due to the limitation of processing technology, the core of the permanent magnet spherical motor (PMS_pM) is generally made of non-conducting magnetic materials. However, the poor magnetic field transfer and concentration effect of the coreless PMS_pM leads to increased energy loss, low torque density, and insufficient driving capability,^4–6 which greatly affects the application prospect of spherical motors.

Soft magnetic composite materials (SMCM) are easy to machine into complex shapes⁷ and the magnetic behavior of SMCM is isotropic,⁸ it has been intensively studied in the field of permanent magnet motors.^9–11 SMCM represent a convenient solution for machines in which the magnetic flux lines run in three dimensional paths. Therefore, SMCM are used to make into the stator core of spherical motors in this paper. However, the introduction of SMCM makes the air gap unevenly distributed in three-dimensional space, which brings multi-dimensional cogging torque and thus causes spherical motor vibration. Reference^12–14 performed analytical solution the cogging torque of conventional cylindrical and linear permanent magnet motors and obtained the constraint relationship between the cogging torque and the structural parameters of the motor. However, the three-dimensional structure of the spherical motor determines that its cogging torque also needs to be analyzed in three-dimensional, conventional methods are no longer suitable. The analytical model for the magnetic field of the PMS_pM with iron core was obtained in References,^15–18 but the cogging torque was not analyzed.

In,^19–22 the cogging torque of a PMS_pM was studied using the finite element analysis (FEA). The FEA has high accuracy in engineering analysis, but it is difficult to provide theoretical support for the optimization direction of structural parameters. Reference²³analyzed the cogging torque of a spherical motor, but the shape of the stator was approximated as a barrel, so the cogging torque was only solved in one dimension. Reference²⁴ solved the multi-dimensional electromagnetic torque in the spherical motor, but it did not analyze the cogging torque separately during the solving process. The air gap magnetic field of the spherical motor with an iron core was analyzed in our previous work,¹⁵ and it found that the air gap magnetic field had severe distortion at different positions of the rotor.

Based on the PMS_pM with SMCM iron core structure proposed in,¹⁵ the analytical model of cogging torque is derived in this paper. Firstly, the distribution functions characterizing the tooth slot structure in the longitude and latitude directions is obtained by the Fourier expansion method. Secondly, the analytical model of air gap magnetic flux density is deduced by combining the distribution function of slot structure and the air gap magnetic flux density function of the slotless structure. Then, the analytical model of the cogging torque in rotation and yaw is obtained by using the virtual work method. Finally, the result was validated on a finite element simulation and torque experiment platform.

Structure and working principle

The structure of the PMS_pM with SMCM core is shown in Fig. 1. The motor consists of a spherical shell stator with teeth and a permanent magnet spherical rotor.

Figure 1.

Structural diagram of PMS_pM.

In the figure, R₁ and R_s are the outer and inner radii of the stator shell, R₂ is the inner stator radius from the stator teeth to the center of the ball, R₃ and R₄ are the outer and inner radii of the rotor, r_s and h_s are the radius and length of the stator teeth, and r_r and h_r are the radius and height of the rotor magnetic pole.

The rotor body is made of resin material by 3D printing technology, and the stator is made of SMCM, which is the Somaloy 1000 5P material from Hoganas, Sweden. Sixteen NdFeB35 permanent magnets are embedded on the surface of the rotor ball, with the upper and lower layers uniformly spaced on the latitude lines of 63.43° and 116.57°. The N and S poles are arranged alternately in two layers. The shape of the permanent magnet is designed as a solid cylinder with parallel magnetization along the column axis. Sixteen stator teeth are mounted uniformly in two layers on the 63.43° and 116.57° latitude lines of the stator. Spherical bearings are mounted on the bottom of the stator to support the rotor, and the spherical rotor can achieve multi-DOF movement in the sphere space. The parameters and symbols of the structure are listed in Table 1.

Table 1.

Simulation model parameters of the PMS_pM with SMCM core.

Symbols	Parameters	Value
R ₁	Stator shell outer radius	58 mm
R ₂	Stator inner radius	36 mm
R _s	Stator shell inner radius	53 mm
R ₃	Rotor outer radius	33.5 mm
R ₄	Rotor inner radius	29.65 mm
r _s	Stator tooth radius	7.5 mm
h _s	Stator tooth length	17 mm
r _r	Rotor magnetic pole radius	7.5 mm
h _r	Rotor magnetic pole height	3 mm
g	Air gap length	2.5 mm
p	Number of poles	8

The schematic diagram of the motion of the PMS_pM is shown in Fig. 2.

Figure 2.

Schematic diagram of the motion of the PMS_pM.

When the stator coil is energized, the magnetic field of the energized coil is made to interact with the magnetic field of the rotor pole, thereby producing electromagnetic torque. When the coils of the same latitude are energized, the rotor is caused to rotate around the Z-axis under the action of the electromagnetic torque. When the coils of the same longitude are energized, the rotor is made to pitch around the X-axis or yaw around the Y-axis. Therefore, the three-DOF motion of the motor can be realized by changing the powering strategy of the stator windings. In the spherical coordinate, the spatial position of any point P is (r, θ, φ), r is the distance between the origin O and point P, θ is the angle between the line segment OP and the positive Z-axis, φ is the angle formed by rotating counterclockwise from the X-axis to OM, and M is the projection of the point P on the XOY plane. The motion of the PMS_pM is described by (δ_r, δ_y, δ_p), δ_r is rotation angle, δ_y is the yaw angle, and δ_p is the pitch angle. ε₁ and ε₂ are the longitude and latitude angle crossed by the magnetic pole.

Analytical model of cogging torque

For the PMS_pM, the utilization of SMCM as the stator core has the potential to enhance the air gap flux density and output torque. However, it unavoidably gives rise to the cogging torque. To analyze the relationship between the cogging torque and the motor parameters, the analytical model of the cogging torque is obtained by using the virtual work method combined with the Fourier expansion method.

Before the cogging torque is discussed, it is necessary to define the stator and rotor coordinate system to describe of the motion of the PMS_pM. The stator coordinate system is a stationary coordinate system, the rotor coordinate system is a motion coordinate system that follows the rotor. The rotor position of a single-DOF motor is determined by the initial position θ₀ and the rotation angle θ_r, and the rotor position can be expressed as θ_t=θ₀+θ_r. The rotor position of the PMS_pM is also related to the initial position and the rotation angle, but the difference is that the rotation angle that determines the rotor position is no longer a single variable. The rotor position is determined by the initial position (δ_r₀,δ_y₀,δ_p₀) and the rotation angle (δ_r,δ_y,δ_p), and the arbitrary rotor position can be expressed as δ_rt=δ_r₀+δ_r, δ_yt=δ_y₀+δ_y, δ_pt=δ_p₀+δ_p. In addition to the rotor position, distribution functions of the magnetic field and the air gap length, in the stator and the rotor coordinate system are described by (θ_d,φ_d) and (θ_z,φ_z), respectively.

Because the yaw and pitch motions are similar, the solution process of yaw cogging torque and pitching cogging torque is the same, so only the solution process of rotation cogging torque and yaw cogging torque is analyzed. The schematic diagrams of the rotation and yaw cogging torque are shown in Fig. 3.

Figure 3.

Schematic diagram of cogging torque direction.

Fig. 3 depicts the direction of rotor motion and direction of cogging torque during rotation and yaw.

For the convenience of research, the following assumptions are made:

The permeability of permanent magnets is the same as vacuum;

The shape of stator slot is simplified as a rectangle;

End effects are not considered;

fundamental in the air gap is considered.

According to the virtual work method, the cogging torque is defined as the negative derivative of the magnetic field energy W to the rotor rotation angle δ.

T_{c o g} = - \frac{\partial W}{\partial δ}

(1)

Since the magnetic permeability of the SMC core is much greater than that of the air gap, it is approximated that the magnetic field energy of the air gap is the total amount of magnetic field capacity.

W \approx W_{g a p} = \frac{1}{2 μ_{0}} \int_{V} B_{r}^{2} (θ_{z}, φ_{z}, δ) d V

(2)

Where, μ₀ represents the vacuum permeability and B_r(θ_z, φ_z, δ) represents the air gap magnetic flux density distribution function with rotor rotation angle.

Characterization function of tooth slot structure

Affected by the tooth slot structure, the distribution function of air gap magnetic flux density with rotor rotation angle can be expressed as:

B_{r} (θ_{z}, φ_{z}, δ_{r}) = B_{r} (θ_{z}, φ_{z}) D (δ_{r}, φ_{z})

(3)

B_{r} (θ_{z}, φ_{z}, δ_{y}) = B_{r} (θ_{z}, φ_{z}) D (δ_{y}, θ_{z})

(4)

Where, B_r(θ_z,φ_z,δ_r) is the magnetic flux density distribution function with rotation angle, B_r(θ_z,φ_z,δ_y) is the magnetic flux density distribution function with yaw angle, and B_r(θ_z,φ_z) is the distribution function of air gap magnetic flux density without tooth slot structure.

D(δ_r,φ_z) and D(δ_y,θ_z) are characterization functions of tooth slot structure in rotation and yaw, they can be expressed as:

D (δ_{r}, φ_{z}) = \frac{h_{m}}{h_{m} + g (δ_{r}, φ_{z})}

(5)

D (δ_{y}, θ_{z}) = \frac{h_{m}}{h_{m} + g (δ_{y}, θ_{z})}

(6)

Where, h_m is the length of the permanent magnet in the direction of magnetization, g(δ_r, φ_z) and g(δ_y, θ_z) are the effective air gap length distribution functions in rotation and yaw, respectively.

The key to deriving the analytical model of the cogging torque is the analysis of B_r(θ_z,φ_z), D(δ_r, φ_z) and D(δ_y, θ_z), the detailed analysis process is shown in Fig. 4. Firstly, according to the boundary conditions and Laplace's equation, the analytical model of air gap magnetic flux density for the slotless structure is established. Then, the distribution function of the effective air gap length is obtained based on the tooth slot structure, and the characterization function of tooth slot structure is obtained. Combining the air gap magnetic flux density for the slotless structure with the characterization function of the tooth slot structure, the air gap magnetic flux density for the tooth slot structure is obtained. It describes accurately the air gap magnetic flux density of the PMS_pM and obtains the magnetic field characteristics, thus optimizing the design to reduce cogging torque.

Figure 4.

Analysis flowchart of analytical model for the PMS_pM.

Firstly, calculate the B_r(θ_z,φ_z). The PMS_pM is divided into five regions according to the magnetic field characteristics of the space as shown in Fig. 5.

Region 1: The air region outside the stator.

Region 2: The stator and stator teeth.

Region 3: The air region between the stator and rotor.

Region 4: The permanent magnets.

Region 5: The rotor yoke.

Figure 5.

Division of solution area.

The Laplace's equation of the scalar magnetic potential for the PMS_pM is expressed as¹⁵:

\frac{1}{r^{2}} [\frac{\partial}{\partial r} (r^{2} Φ_{i}) + \frac{1}{\sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial Φ_{i}}{\partial θ}) + \frac{1}{\sin^{2} θ} \frac{\partial Φ_{i}}{\partial φ}] = 0

(7)

The boundary conditions between the five regions can be expressed as:

{\begin{matrix} {B_{i r} |}_{r = R_{i}} = {B_{(i + 1) r} |}_{r = R_{i}}, (i = 1, 2, 3, 4) \\ {H_{i θ} |}_{r = R_{i}} = {H_{(i + 1) θ} |}_{r = R_{i}}, (i = 1, 2, 3, 4) \\ {H_{i φ} |}_{r = R_{i}} = {H_{(i + 1) φ} |}_{r = R_{i}}, (i = 1, 2, 3, 4) \\ \begin{matrix} Φ_{1} |_{r \to \infty} = 0 \\ Φ_{5} |_{r \to 0} \neq \infty \end{matrix} \end{matrix}

(8)

The magnetic field in space is solved by the Laplace's equation and the scalar magnetic potential is represent by (r, θ_z, φ_z).

Φ_{i} = \sum_{l = 0}^{\infty} \sum_{m = - l}^{l} [A_{l, i}^{m} r^{l} + B_{l, i}^{m} r^{- (l + 1)}] [Y_{l}^{m} (θ_{z}, φ_{z})]

(9)

Where, i = 1,2,3,4,5, represents the magnetic field solution region. A^m_l,.i and B^m_l,i are the coefficients to be determined from the boundary conditions, Y^m_l (θ_z, φ_z) is the spherical harmonic function, l represents the degree, and m represents the order.

Y^m_l (θ_z, φ_z) can be expressed as:

Y_{l}^{m} (θ_{z}, φ_{z}) = \sqrt{\frac{2 l + 1}{4 π} \frac{(l - m)!}{(l + m)!}} P_{l}^{m} (\cos θ_{z}) e^{j m φ_{z}}

(10)

Where, P^m_l (cosθ_z) is the differential form of the associated Legendre function, which can be expressed as:

P_{l}^{m} (x) = \frac{1}{2^{l} l!} (1 - x^{2})^{\frac{m}{2}} \frac{d^{m + l}}{d x^{m + l}} (x^{2} - 1)^{l}, x = \cos θ_{z}

(11)

Because only the radial component of the air gap flux density can generate torque, the radial fundamental wave magnetic flux density of the air gap obtained by spherical harmonic expansion can be expressed as:

B_{r} (θ_{z}, φ_{z}) = - μ_{0} [(5 A_{5, 3}^{- 4} r^{4} - 6 B_{5, 3}^{- 4} r^{- 7}) Y_{5}^{- 4} + (5 A_{5, 3}^{4} r^{4} - 6 B_{5, 3}^{4} r^{- 7}) Y_{5}^{4}]

(12)

Where, A^-4_5,3, B^-4_5,3, A⁴_5,3 and B⁴_5,3 are the scalar magnetic potential coefficients, Y-4 5and Y4 5are the spherical harmonic functions abbreviated respectively.

Analytical model of cogging torque in rotation dimension

The relative position between the permanent magnet and the stator teeth in rotation is shown in Fig. 6.

Figure 6.

Relative position of stator and rotor in rotation.

The relative positions of rotor pole PM2 and stator tooth S2 are taken as an example. δ_r₀ represents the initial position of the rotor, δ_r represents the rotation angle, and δ_rt represents the position after rotation. φ_d and φ_z describe the distribution of the air gap length in the stator and rotor coordinate system, and the relative position relationship is φ_d = φ_z + δ_r. φ_d = 0° represents the centerline of the stator tooth S2, and φ_z = 0° represents the centerline of the magnetic pole PM2.

The distribution function of effective air gap length in the stator coordinate system is obtained according to the geometric structure of the air gap, as shown in Fig. 7.

Figure 7.

Distribution function of effective air gap length in rotation.

As shown in Fig. 7, it assumed that the magnetic flux path crosses the air gap and magnet region in the radial direction, and enters the slot opening with a circular trajectory, the distribution function of effective air gap length in the stator coordinate system in rotation is expressed as:

g (φ_{d}) = {\begin{matrix} g \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} & 0 \leq φ_{d} \leq φ_{α} \end{matrix} \\ \begin{matrix} g + \frac{π b_{r 0}}{2 φ_{0}} (φ_{d} - φ_{α}) & φ_{α} \leq φ_{d} \leq \frac{φ_{1}}{2} \end{matrix} \end{matrix}

(13)

Where, b_r₀ is the width of the slot opening in rotation, φ_α=(φ₁-φ₀)/2, φ₁ and φ₀ are the slot pitch and slot opening, respectively, and g is the air gap length.

Combining φ_d=φ_z+δ_r and (5), D(δ_r, φ_z) can be expressed as:

D (δ_{r}, φ_{z}) = D (φ_{d}) = \frac{h_{m}}{h_{m} + g (φ_{d})}

(14)

Where, D(δ_r, φ_z) and D(φ_d) represent the characterization functions of the tooth slot structure in the rotor coordinate system and stator coordinate system, respectively.

To simplify the calculation, (15) can be obtained by Fourier expansion.

D^{2} (δ_{r}, φ_{z}) = G_{r 0} + \sum_{n = 1}^{\infty} G_{r n} \cos n z_{r} (δ_{r} + φ_{z})

(15)

Where, z_r represents the number of stator teeth, G_r₀ and G_rn represent the Fourier expansion coefficients, and the expression is:

{\begin{matrix} G_{r 0} = \frac{2}{φ_{1} / 2} \int_{0}^{\frac{φ_{1}}{2}} D^{2} (φ_{d}) d φ_{d} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \\ G_{r n} = \frac{2}{φ_{1} / 2} \int_{0}^{\frac{φ_{1}}{2}} D^{2} (φ_{d}) \cos \frac{2 n π φ_{d}}{φ_{1}} d φ_{d} \end{matrix}

(16)

Combining (1), (2), (3), (12), (15) and (16), the analytical expression of cogging torque in rotation is given.

\begin{aligned} T_{c o g r} = - \frac{1}{2 μ_{0}} \frac{\partial}{\partial δ_{r}} [\int_{θ_{a}}^{θ_{b}} \int_{0}^{2 π} \int_{R_{3}}^{R_{2}} \\ B_{r}^{2} (θ_{z}, φ_{z}) D^{2} (δ_{r}, φ_{z}) r^{2} \sin θ d r d φ d θ] \end{aligned}

(17)

Where, θ_a and θ_b are the latitudes of the upper and lower layers of the magnetic poles, R₃ and R₂ are the radii of the inner and outer spheres of the air gap.

The cogging torque T_cogr in rotation is related to the distributions of D²(δ_r, φ_z) and B_r²(θ_z, φ_z). B_r²(θ_z, φ_z) is related to the rotor magnetic poles, with a period of 2p = 8. D²(δ_r, φ_z) is related to the stator teeth, with a period of z_r = 8. B_r(θ_z, φ_z) is a series of cosine waves with a period of 2kp in the direction of φ_z, when it is expanded by the spherical harmonic function. D²(δ_r, φ_z) is a series of cosine waves with period nz_r when it is expanded by Fourier series expansion. Therefore, B_r²(θ_z, φ_z) and D²(δ_r, φ_z) can generate cogging torque only when 2kp = nz_r (k, nz_r are positive integers). Because the cogging torque is solved using the fundamental wave component of the air gap radial magnetic flux density in the φ_z direction, it can get k = 1, which results in n = 1, so the cogging torque model in rotation in this paper is based on the fundamental waves of B_r²(θ_z, φ_z) and D²(δ_r, φ_z).

Analytical model of cogging torque in yaw dimension

The relative position between the permanent magnet and the stator teeth in rotation is shown in Fig. 8.

Figure 8.

Relative position of the stator and rotor in yaw.

The relative positions of rotor pole PM1 and stator tooth S1 are taken as an example. δ_y₀ represents the initial position of the rotor, δ_y represents the yaw angle, δ_yt represents the position after yaw, θ_d and θ_z describe the distribution of the air gap length in the stator and rotor coordinate system, and the relative position relationship is θ_d=θ_z+δ_y. θ_d=θ_du represents the centerline of the stator tooth S1, and θ_z=θ_zu represents the centerline of the magnetic pole PM1, θ_du=θ_zu = 63.43°.

B_r(θ_z, φ_z) is the same as in the previous section, and the distribution function of effective air gap length in yaw is obtained according to the geometrical structure of the air gap, as shown in Fig. 9.

Figure 9.

Distribution function of effective air gap length of yaw.

As shown in Fig. 9, again assuming that the magnetic flux path passes crosses the air gap and magnet region in the radial direction, enters the slot opening with a circular trajectory, the distribution function of effective air gap length in yaw is expressed as:

g (θ_{d}) = {\begin{matrix} g \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} & 0 \leq θ_{d} \leq θ_{β} \end{matrix} \\ g + \frac{π b_{y 0}}{2 θ_{0}} (θ_{d} - θ_{β}) \begin{matrix} θ_{β} \leq θ_{d} \leq \frac{θ_{1}}{2} \end{matrix} \end{matrix}

(18)

Where, b_y₀ is the width of the slot opening in yaw, θ_β=(θ₁-θ₀)/2, θ₁ and θ₀ are the slot pitch and slot opening, respectively, and g is the air gap length.

Combining θ_d=θ_z+δ_y and (6), D(δ_y, θ_z) can be expressed as follows:

D (δ_{y}, θ_{z}) = D (θ_{d}) = \frac{h_{m}}{h_{m} + g (θ_{d})}

(19)

Where, D(δ_y, θ_z) and D(θ_d) are the characterization functions of the tooth slot structure in the rotor coordinate system and stator coordinate system, respectively.

To simplify the calculation, (20) can be obtained by the Fourier expansion.

D^{2} (δ_{y}, θ_{z}) = G_{y 0} + \sum_{n = 1}^{\infty} G_{y n} \cos n z_{y} (δ_{y} + θ_{z})

(20)

Where, z_y represents the number of stator teeth, G_y₀ and G_yn represent the Fourier expansion coefficients, and the expression is:

{\begin{matrix} \begin{matrix} G_{y 0} = \frac{2}{θ_{1} / 2} \int_{0}^{\frac{θ_{1}}{2}} D^{2} (θ_{d}) d θ_{d} & \begin{matrix}  \end{matrix} \end{matrix} \\ G_{y n} = \frac{2}{θ_{1} / 2} \int_{0}^{\frac{θ_{1}}{2}} D^{2} (θ_{d}) \cos \frac{2 n π θ_{d}}{θ_{1}} d θ_{d} \end{matrix}

(21)

Combining (1), (2), (4), (12), (20) and (21), the analytical expression for cogging torque in yaw is given.

\begin{aligned} T_{c o g y} = - \frac{1}{2 μ_{0}} \frac{\partial}{\partial δ_{y}} [\int_{φ_{a}}^{φ_{b}} \int_{0}^{π} \int_{R_{3}}^{R_{2}} \\ B_{r}^{2} (θ_{z}, φ_{z}) D^{2} (δ_{y}, θ_{z}) r^{2} \sin θ d r d θ d φ] \end{aligned}

(22)

Where, φ_a and φ_b are the longitudes of the left and right edges of the magnetic poles, and R3 and R2 are the radii of the inner and outer sphere of the air gap.

The cogging torque T_cogy in yaw is related to the distributions of B_r²(θ_z, φ_z) and D²(δ_y, θ_z). B_r²(θ_z, φ_z) is related to the rotor magnetic poles, with a period of 2p = 2. D²(δ_y,θ_z) is related to the stator teeth, with a period of z_y = 2. B_r(θ_z,φ_z) is a series of cosine waves with a period of 2kp in the direction of θ_z, when it is expanded by the spherical harmonic function. D²(δ_y,θ_z) is a series of cosine waves with period nz_y, when it is expanded by Fourier series expansion. Thus B_r²(θ_z, φ_z) and D²(δ_y, θ_z) can generate cogging torque only when 2kp = nz_y(k, z_y are positive integers). Becausethe cogging torque is solved using the fundamental wave component of the air gap radial magnetic flux density in the θ_z direction, it can get k = 1, which results in n = 1. The cogging torque model in yaw in this paper is based on the fundamental waves of B_r²(θ_z, φ_z) and D²(δ_y, θ_z).

FEA validation of the analytical cogging torque model

The analytical model is compared with the FEA results to verify the accuracy of the analytical model of the cogging torque. The simulation setup information table is shown in Table 2.

Table 2.

PMS_pM with SMCM core simulation model parameters.

Setting information	Simulation parameters
3D static magnetic field	Simulation environment
Direct drawing modeling	3D model
Zero Tangential component	Boundary condition
Adaptive meshing	Meshing
Default system settings	Analysis setting
Air gap magnetic flux density	Analysis indicator

The analytical results and FEA results are shown in Fig. 10. The amplitudes of rotation and yaw cogging torque in FEA are 0.3384N·m and 0.3142N·m. The rotation cogging torque is a sinusoidal waveform with a cycle of 45°, and the yaw cogging torque is a central symmetric sinusoidal waveform centered at δ_y = 0°. When the yaw angle is greater than 25°, the error between the analytical result and simulated result of yaw cogging torque is relatively large. That is caused by two reasons. Firstly, the magnetic field at the end of the PMS_pM is complex, but the effective air gap length function is simplified. Secondly, it only considers the fundamental component of the air gap magnetic flux density and the fundamental component of the Fourier decomposition of effective air gap length function, without adding actual harmonics.

Figure 10.

Comparison of analytical model and FEA of cogging torque. (a) Cogging torque in rotation (b) Cogging torque in yaw.

The relative error is used to assess the accuracy of the analytical calculations of the rotation and yaw cogging torque, and the relative error calculation formulation is as follows:

{\begin{matrix} u_{r} = \frac{1}{N} \sum_{i = 0}^{N} | T_{r F E M} - T_{c o g r} | / T_{r F E M} \\ u_{y} = \frac{1}{N} \sum_{i = 0}^{N} | T_{y F E M} - T_{c o g y} | / T_{y F E M} \end{matrix}

(23)

Where, T_rFEM and T_yFEM are the finite element simulation values, T_cogr and T_cogy are the analytical values, u_r and u_y are the average relative errors between the analytical model and the FEA results of the rotation cogging torque and yaw cogging torque, and N pole is the number of numerical points.

The magnitude and average relative error are shown in Table 3, and the errors are within the allowable range of engineering.

Table 3.

Analysis of cogging torque error

Cogging torque	Rotation	Yaw
FEA amplitude	0.3384N·m	0.3142N·m
Analytical amplitude	0.3157N·m	0.3204N·m
Mean relative	6.88%	8.35%

The effect of motor parameters on cogging torque in analytical model

The constraint relationship between physical quantities and the structural parameters of the spherical motor is established by the analytical model, which can easily analyze the influence of the structural parameters of the motor and has guiding significance for the structural optimization of the motor. Without changing the overall structure of the spherical motor with SMCM core, analyze the relationship between the permanent magnet thickness h_m, air gap length g, stator slot opening angle φ₀ and the cogging torque in the analytical.

The values of motor parameters h_m, g and φ₀ are changed in the analytical model, and h_m is designed to be a step of 0.5 mm with a variation range of 2mm-4 mm, g is designed to be a step of 0.2 mm with a variation range of 2.1mm-2.9 mm, and φ₀ is designed to be a step of 2° with a variation range of 16.5°-24.5°. Through analytical calculation, the results are shown in Fig. 11.

Figure 11.

The effect of motor parameters on cogging torque. (a) Rotation cogging torque (b) Yaw cogging torque.

In the above figure, the size and color of the circles indicate the rotation and yaw cogging torque, respectively. When the thickness of the permanent magnet h_m increases, the cogging torque will increase, and the rate of the cogging torque increase is uniform; when the air gap length g increases, the cogging torque will decrease, and the rate of the cogging torque decrease is also uniform; when the opening angle of the stator slot φ₀ increases, the rotation cogging torque will increase, and the rate is slow, and the yaw cogging torque is almost unaffected.

According to the effect of the structural parameters of the motor on the cogging torque, the weakening of the cogging torque can be considered from the aspects of decreasing the thickness of the permanent magnets, increasing the air gap length, and decreasing the opening angle of the stator slot.

The air gap magnetic flux density will increase with the increase of permanent magnet thickness, but there will be a saturation point. The increase of air gap length will make the air gap magnetic flux density decrease, but too small air gap length requires high requirements for machining and assembly process. Based on the above analysis, a set of optimal design parameter combinations are selected as h_m = 2 mm, g = 2.9 mm, φ₀ = 16.5°. The results of the cogging torque analysis before and after the optimization are compared, as shown in Fig. 12.

Figure 12.

Comparison of analytical model and FEA for cogging torque. (a) Rotation cogging torque (b) Yaw cogging torque.

As shown in the above figure, the magnitudes of the analytical results of the rotation cogging torque before and after optimization are 0.3157N·m and 0.1272N·m, respectively, which is weakened by 59.71%. The magnitude of the analytical results of the yaw cogging torque are 0.3204N·m and 0.1462N·m, respectively, which is weakened by 54.37%. The relationship between the structural parameters and the cogging torque can be quickly and accurately reflected by the analytical model. A lot of time is saved compared with the FEA, and the foundation for the structural optimization of the PMS_pM with SMCM core is laid by the analytical model of the cogging torque.

Experimental validations

Experimental measurements were conducted on the rotation and yaw cogging torque to verify the correctness of the cogging torque analytical model. The torque measurement platform is shown in Fig. 13. When measuring the rotation torque, the sensor support plate is fixed in the XOZ plane, and the rotation torque is applied to the pressure sensor through the torque output guide shaft and the horizontal output rod. When measuring the yaw torque, the sensor support plate moves in the YOZ plane through the base slide to follow the translation of the horizontal output rod, thus obtaining the measured value of yaw torque. During the experiment, the rotor of the motor was made to rotate and yaw in the planned direction by an external force, so as to get the values of the cogging torque in rotation and yaw at the set measurement nodes.

Figure 13.

Torque measurement platform of the PMS_pM with SMCM core.

The experimental data of the rotation cogging torque and yaw cogging torque are obtained with a measurement step of 5°, and the analyzed results and experimentally measured waveforms are shown in Fig. 14.

Figure 14.

Experimental and analytical results of cogging torque. (a) Rotation cogging torque (b) Yaw cogging torque.

It can be seen that, the analytical model and experimental measurement amplitude of the rotation cogging torque of the PMS_pM with SMCM core are 0.3157N·m and 0.3308N·m, respectively, and the analytical model and experimental measurement amplitude of the yaw cogging torque are 0.3204N·m and 0.3150N·m, respectively. The rotation cogging torque is a periodical variation with a period of 45°. The experimental results are consistent with the analytical analysis. When the rotor rotates from the initial position δ_r = 0° to the positive direction, it will be subjected to the action of the negative direction of the cogging torque, which hinders the rotation motion. It is in the unstable equilibrium state when the rotor rotates to δ_r = 22.5°. The yaw cogging torque is a centrosymmetric change, and the center of symmetry is δ_y = 0°. When the rotor is yawing from the initial position δ_y = 0° to the positive direction, it will be acted by the cogging torque in the negative direction to hinder the yawing motion. On the contrary, it holds true. Neglecting factors such as friction of the support structure, the experimental data of the rotation and yaw cogging torque are in good agreement with the analytical results.

Equation (24) is utilized to calculate the relative error between the experimental measurements and the analytical results, and the relative error of the rotation cogging torque and yaw cogging torque data is shown in Fig. 15.

relative error = (T_{A n a} - T_{E x p}) / T_{A n a}

(24)

Figure 15.

Relative error between the experimental measurements and the analytical results. (a) Rotation cogging torque (b) Yaw cogging torque.

Where, T_Ana is the analytical result, and T_Exp is the experimental result.

From Fig. 15, it can be seen that the average values of the relative errors between the experimental and analytical results of the rotation and yaw cogging torque are 6.09% and 7.56%, respectively. Relatively large errors manifest in the vicinity of 0 N·m, and this portion of the relative error fails to measure the overall accuracy level. Therefore, the error data in this part is disregarded. Given the objective existence of mechanical errors sensor measurement errors and other errors in the experimental measuring device, relative errors are within acceptable limits.

Conclusion

Firstly, in this paper, the analytical models of the rotation and yaw cogging torque for the double-layer eight-pole PMS_pM with SMCM core are derived by using the Fourier expansion method and the virtual work method. Moreover, the models are verified through finite element analysis. Then, the relationships between the permanent magnet thickness h_m, air gap length g, stator slot opening angle φ₀ and the cogging torque of the motor are discussed based on the analytical model. Finally, the analytical model of cogging torque is verified by experiments. The research work on cogging torque in this paper lays a theoretical foundation for the structural optimization design of the PMS_pM with iron core.

Footnotes

Acknowledgements

This work was supported by the Department of National Natural Science Foundation of China (52377065).

ORCID iD

Xinyuan Zhang

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

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