Analysis of Magnetic Field and Torque Features of Improved Permanent Magnet Rotor Deflection Type Three-Degree-of-Freedom Motor

Abstract

This paper proposes a novel layered permanent magnet motor (N-LPM), which is based on a three-degree-of-freedom (3-DOF) permanent magnet motor. Compared with the former, the improved N-LPM air gap magnetic density, torque and structure stability have been significantly improved. The proposed N-LPM has three layers of stator along the axis direction, and each layer of stator has three-phase winding. In order to calculate the magnetic field and torque distribution of the N-LPM, an analytical method (AM) is proposed. For performance verification and accurate calculation, finite-element analysis (FEA) is adopted. The two kinds of motors before and after the improvement are compared, and their magnetic field, torque and stability are analyzed. The optimization rate is defined to evaluate the performance of the motor before and after improvement. The results show that the radial flux density, rotation torque, deflection torque and the volume optimization rate of the permanent magnet of the improved motor are 80%, 25%, 50% and 54.72% respectively, and the comprehensive performance is improved significantly.

1. Introduction

The traditional multi-degree-of-freedom (multi-DOF) motor is composed of a large number of traditional motors and corresponding mechanical links, but the combination of these structures is complex and cumbersome, and difficult to maintain, so it is difficult to apply them to fine motion occasions [1,2]. Multi-DOF motors can be widely used in machine tools, mechanical arms and other fields [3,4]. With the development of science and technology and the progress of manufacturing technology, many motors with multiple degrees-of-freedom have be designed, tested and studied. Although the multi-degree-of-freedom motor model with coil excitation is cheaper, the introduction of an excitation mechanism will lead to a larger overall model of the motor, which is difficult to be used in small applications such as robot joints [5,6]. The volume of multi-DOF motors using a permanent magnet as rotor will be improved obviously, but the corresponding cost will also be increased. Especially when some permanent magnet materials with higher shape requirements are used, the cost will be higher [7,8]. Partial permanent magnet multi-degree-of-freedom motors have very low slot utilization rates. This will not only affect the magnetic density distribution and structural stability of the motor, but also make the temperature of some stator coils rise very high. When the temperature of the permanent magnet is high, it is easy to produce a demagnetization effect, which will damage the equipment if it is used for a long time [9,10,11]. In view of the disadvantages of the existing multi-degree-of-freedom motor, such as high cost, low air gap magnetic density and unstable structure and slot utilization, a novel new layered permanent magnet motor (N-LPM) is proposed based on the permanent magnet deflection three-degree-of-freedom motor [12]. The N-LPM has the advantages of small volume, high slot utilization, high magnetic density amplitude and mechanism stability, which make up for the shortcomings of the existing multi-degree-of-freedom motor. At the same time, the N-LPM also has the characteristics of low flux leakage and high torque, which is difficult to achieve in the existing multi-degree-of-freedom motor.

In this paper, a 3-DOF motor model with permanent magnet deflection type rotor is improved. A more compact model is proposed for the stator part of the motor. The purpose is to reduce the magnetic leakage as much as possible to make the motor structure more fit for 3-DOF motions. At the same time, the slot space factor can be controlled reasonably, and the current density of the motor will not be affected. The two-layer stator magnetic field path which controls the deflection of the motor tilted to a certain extent and the improved model are analyzed. Analysis of the results shows that the deflection angle of the motor is improved and the deflection torque is also improved by adopting the improved stator model. For the rotor part of the motor, a new type of butterfly permanent magnet is proposed, which can reduce the amount of permanent magnet used and increase the amplitude of the original magnetic field to a large extent. Therefore, the torque is significantly enhanced, and the magnetic resistance between the permanent magnets is reduced, making the magnetic field density more intensive. The model adopts the combination mode of block splicing on the rotor and the more compact winding connection on the stator. The improved structure is compared with the original structure using the analytical method. At the same time, the improved and original models are analyzed using the finite element method. The results show that the magnetic field performance and torque characteristics of the new structure are effectively improved, and the magnetic flux leakage is obviously suppressed.

2. Motor Structure

Figure 1 shows the motor model of the permanent magnet rotor deflection type3-DOF motor, the upper half part of Figure 1 shows the general drawing of the permanent magnet deflection 3-DOF motor, and the next part shows the breakdown assembly drawing of the motor and explains each part to facilitate understanding of the internal structure of the motor. There are three layers of stator in the permanent magnet rotor deflection type 3-DOF motor. The 1st or 3rd layer is mainly responsible for driving the motor to complete the deflection movement, the 2nd layer is responsible for driving the motor to complete the spin rotation movement, and the three layers of stator cooperate with each other to achieve the 3-DOF movement in space. The 2nd layer stator is made of laminated silicon steel sheet. When the motor is driven by AC current, the eddy current loss can be effectively reduced. The 1st or 3rd layer is made of electrical pure iron. On the one hand, it can effectively reduce the magnetoresistance of the stator when deflecting. The frequency of electrified alternating current required for deflection is low. The use of electrical pure iron can meet the needs of movement while saving costs.

Figure 2 shows the eddy current distribution of the stator in different layers when silicon steel sheet material is used, in which the electric signal driven by rotation is input from the 2nd coil, and the electric signal driven by deflection is input from the 1st or the 3rd. It can be seen from Figure 2 that the eddy current loss of the 1st or the 3rd layer is obviously less than that of the 2nd layer, and the current of the 1st or the 3rd layer is only a very small part of the AC component, so the 1st or the 3rd layer using the silicon steel sheet is not necessarily.

The unit mass power consumption of eddy current generated in the metal shell or wire is as follows [13]:

$$P=\frac{{\pi }^2{B}_{\max }^2{D}^2{f}^2}{6k\rho D},$$

(1)

where k is constant, metal shell is 1 and wire is 2. P is power consumption per unit mass. B_max is the peak value of the magnetic field. f is the frequency at which the magnetic field changes, $\rho$ is the resistivity, and D is the density of the material.

According to Formula (1), the eddy current loss curves generated by the two stator coils in Figure 3 can be obtained. Figure 3a shows the eddy current generated by the middle stator coil, and Figure 3b shows the eddy current generated by the Upper and lower stator coils. After comparison, it is found that the eddy current loss analysis method in the rotating layer is basically consistent with the finite element method, and there is a certain deviation between the eddy current loss analysis method in the deflecting layer and the finite element method; because of the difference caused by the low orders of magnitude, the error range is less than 10 × 10⁻³². The analytical method of eddy current loss for deflection layer is basically consistent with the finite element method.

If the frequency of magnetic field is relatively high, the skin effect should also be considered [14]:

$$j\left(z\right)=j_0{\mathbf{e}}^{-z/\delta },$$

(2)

where j₀ is the current density of the conductor surface and δ is the skin depth.

Here, the eddy current loss is only taken as an example and it is not in a degree of concern, so the skin effect does not need to be considered either.

Figure 4 shows the motion model of the 3-DOF motor. Figure 4a shows the rotation motion model of the 3-DOF motor, and Figure 4b shows the deflection motion model of the 3-DOF motor. The two models show the two motion states of the motor.

Figure 5 shows the entity diagram of the motor, and its internal structure has been described in detail in the former figures. In this figure, the drag motor is used to test the back Electron Magnetic Field (EMF) of the motor.

3. Analysis of Stator Model before and after Improvement

In this paper, the analysis of the motor is carried out using ANSYS Maxwell.

ANSYS electronics desktop 2019 software is a large-scale general finite element analysis (FEA) software developed by ANSYS company in the United States. It is the fastest-growing Computer-Aided Engineering (CAE) software in the world. It originated in Pittsburgh, Pennsylvania, the United States.

The basic situation of the magnetic field and torque of N-LPM can be obtained using the finite element software Electronics desktop 2019

Figure 6 shows the improved motor model, and the modification of the improved permanent magnet rotor deflection type 3-DOF motor is mainly composed of stator and rotor parts. The stator part and rotor part are respectively introduced as follows.

3.1. Introduction to Improved Stator

As shown in Figure 6, the comparison diagram of the stator of the motor before and after improvement shows that the slot space factor of the original motor model is lower, and the current density is increased. When the motor is in a long-term operation state, the magnetic resistance and iron loss of the magnetic circuit and magnetic field waveform can be with great promotion.

In the figure, l_h is the length of the stator teeth, l_r is the width of the stator teeth, l_y is the thickness of the stator yoke, $\chi$ is the air gap span, ${\chi }_c$ is the slot span, ${\chi }_t$ is the tooth span, ${\chi }_h$ is the vertical span of the 1st or 3rd layer stator.

3.2. Improved Magnetic Field Flux Line Comparison

The comparison before and after the stator improvement of the permanent magnet rotor deflection type 3-DOF motor is shown in Figure 6. It can be seen that the utilization rate between stator slots of the motor has been significantly improved, that is, the slot rate can be expressed as:

$$S_f=\frac{N_{i1}{N}_{s1}{D}^2}{A_{ef}}$$

(3)

where d is the conductor diameter, N_i1 and N_S1 are the winding turns and parallel winding number respectively, and A_ef is the effective area of slot after subtracting the insulation area.

Figure 7 shows the distribution diagram of the magnetic field flux lines when the permanent magnets’ lower layer coils are normally energized. It can be seen from Figure 7 that the magnetic field flux lines of each winding part have been significantly improved after the optimization of the stator. Especially in the air gap part, the density of the magnetic field lines at the air gap of the improved stator structure are obviously much greater than those of the stator structure before the improvement, which can be verified by the magnetic vector.

In rectangular coordinate system, the relationship between the magnetic vector and current density (Poisson equation) can be expressed as follows [15]:

$$A=\frac{\mu }{4\pi }{\displaystyle \underset{V}{\int }\frac{JdV}{R}}$$

(4)

where J is current density.

The distribution of the magnetic field flux lines of the central section of the motor through the magnetic vector is shown in Figure 7. It can be seen from the figure that when the original stator is used, the distribution of the magnetic field flux lines in the air gap except the pole shoe is sparse, close to 0, while the distribution of the magnetic field flux lines in the pole shoe is gradually compact. When the improved stator is used, the amplitude between the air gaps of the motor is significantly increased, and the magnetic field flux lines in the pole shoe are significantly increased. The distribution of the magnetic field flux lines in Figure 8 is basically consistent with the trend and amplitude of the magnetic field flux lines in the air gap obtained using the finite element method in Figure 7, which verifies the accuracy of the analytical method.

The optimization effect of stator improvement on magnetic field flux lines can be quantitatively expressed as follows:

$$\Delta S={\displaystyle {\int }_0^{2\pi }abs\left({\xi }_{NFL}\right)}d\phi -{\displaystyle {\int }_0^{2\pi }abs\left({\xi }_{OFL}\right)}d\phi ={\displaystyle {\int }_0^{2\pi }\left[abs\left({\xi }_{NFL}\right)-abs\left({\xi }_{OFL}\right)\right]}d\phi$$

(5)

where $abs({\xi }_{NFL})$ is the absolute value of the improved stator flux line distribution function when r = 15.5 mm, and $abs({\xi }_{OFL})$ represents the absolute value of the improved stator flux line distribution function when r = 15.5 mm.

The result of Formula (5) is the change of the distribution of the magnetic field flux lines before and after the change of the stator. By comparing Formula (5) with the absolute value area of the distribution function of the magnetic field flux lines before and after the improvement, the optimization rate of the magnetic field flux lines before and after the improvement can be obtained. The optimization rate of the magnetic field lines can be defined as

$${\eta }_{FLS}=\frac{\Delta S}{{\displaystyle {\int }_0^{2\pi }abs\left[{\xi }_{OFL}\right]}d\phi }\times 100\%$$

(6)

Finally, the optimization rate of the magnetic field line can be obtained. The value is regular, which means that the improved magnetic field line is optimized. The larger the value is, the more obvious the optimization degree is. On the contrary, if the value is negative, it means that the magnetic field flux line is weakened after improvement, and the larger the absolute value is, the more obvious the weakening degree is. After calculation, η = 9.1638% is a positive number, and it can be approximately explained that the improved stator can optimize the magnetic line of force by 9.16381% in the air gap under this section. If this concept is raised to three-dimensional space, the volume difference function can be obtained:

$$\Delta V={\displaystyle \underset{\Sigma }{\iint }abs\left({\zeta }_{NFL}\right)}d\phi \mathrm{d}\theta -{\displaystyle \underset{\Sigma }{\iint }abs\left({\zeta }_{OFL}\right)}d\phi \mathrm{d}\theta ={\displaystyle \underset{\Sigma }{\iint }\left[abs\left({\zeta }_{NFL}\right)-abs\left({\zeta }_{OFL}\right)\right]}d\phi \mathrm{d}\theta$$

(7)

where ${\zeta }_{NFL}$ is the absolute value of the three-dimensional distribution function of the improved stator flux line, $abs\left({\zeta }_{OFL}\right)$ is the absolute value of the three-dimensional distribution function of the improved stator flux line, and Σ represents the integral boundary of the flux line.

The same analogy can get the optimization effect of magnetic field lines:

$${\eta }_{FLV}=\frac{\Delta V}{{\displaystyle \underset{\Sigma }{\iint }abs\left({\zeta }_{OFL}\right)}d\phi d\theta }\times 100\%$$

(8)

From Equation (8), η = 22.732% can be obtained. Obviously, the improved stator structure has a significant optimization effect on the magnetic field produced by the coil in the air gap.

3.3. Stator Magnetic Field Comparison

For the 3-DOF motor, the magnetic field can be divided into two parts: air gap rotating magnetic field and air gap deflecting magnetic field [16]. Similar to the traditional motor, the stator air gap magnetic field interacts with the rotor air gap magnetic field, resulting in the corresponding torque to drive the motion of the permanent magnet rotor deflection type 3-DOF motor. The rotating magnetic field and the deflection magnetic field are generated by energizing corresponding coils, wherein the stator rotating current can be expressed as

$$\{\begin{array}{l}{i}_A=I_r\cos \left[\omega t+p\left({\theta }_{r0}+{\theta }_{sr}\right)\right]\\ i_B=I_r\cos \left[\omega t+p\left({\theta }_{r0}+{\theta }_{sr}\right)-\frac{2\pi }{3}\right]\\ i_C=I_r\cos \left[\omega t+p\left({\theta }_{r0}+{\theta }_{sr}\right)+\frac{2\pi }{3}\right]\end{array}$$

(9)

where I_r is peak value phase current,$\omega$ is the electrical frequency, p is the number of pole pairs of rotor, ${\theta }_{r0}$ is the initial rotor position angle, and ${\theta }_{sr}$ is the initial mechanical angle of the rotor.

The rotating magnetic field produced by the stator can be expressed as

$$B_{sr}\left(\theta ,t\right)=B_{s\max }\cos \left[p\left(\theta -{\theta }_{sr}\right)-\omega t\right]$$

(10)

where B_smax is the peak magnetic density of the stator rotating magnetic field, and θ is the mechanical angle.

The stator deflection current can be expressed as

$$\{\begin{array}{l}{i}_{At}=I_t\cos \left(\omega t+p{\theta }_{r0}\right)\\ i_{Bt}=I_t\cos \left(\omega t+p{\theta }_{r0}-\frac{2\pi }{3}\right)\\ i_{Ct}=I_t\cos \left(\omega t+p{\theta }_{r0}+\frac{2\pi }{3}\right)\end{array}$$

(11)

where I_t is the peak value of deflection current.

The deflection magnetic field generated by the stator can be expressed as

$$B_{st}\left(\theta ,t\right)=B_{st\max }\cos \left[\left(p\pm 1\right)\theta -\omega t\right]$$

(12)

where B_stmax is the peak magnetic density of the stator deflection magnetic field.

3.4. Study on Stability of Improved Stator Structure

Considering the stator structure as an elastomer, the degrees-of-freedom can be divided into internal degrees-of-freedom and interface degrees-of-freedom. The differential equation of motion in physical coordinates can be expressed as follows [17]:

$$M_{\mathrm{s}}{\stackrel{..}{u}}_{\mathrm{s}}+C_{\mathrm{s}}{\stackrel{.}{u}}_{\mathrm{s}}+K_{\mathrm{s}}{u}_{\mathrm{s}}=P_{\mathrm{s}}$$

(13)

where

$$\begin{gathered} M_{\mathrm{s}}=\left[\begin{array}{cccc}{\mathrm{M}}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{i}}& {\mathrm{M}}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{j}}& 0& 0\\ {\mathrm{M}}_{\mathrm{s}\mathrm{x}}^{\mathrm{j}\mathrm{i}}& {\mathrm{M}}_{\mathrm{s}\mathrm{x}}^{\mathrm{j}\mathrm{j}}& 0& 0\\ 0& 0& {\mathrm{M}}_{\mathrm{s}\mathrm{y}}^{\mathrm{i}\mathrm{i}}& {\mathrm{M}}_{\mathrm{s}\mathrm{y}}^{\mathrm{i}\mathrm{j}}\\ 0& 0& {\mathrm{M}}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{i}}& {\mathrm{M}}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{j}}\end{array}\right];{\mathrm{C}}_{\mathrm{s}}=\left[\begin{array}{cccc}{\mathrm{C}}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{i}}& {\mathrm{C}}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{j}}& 0& 0\\ {\mathrm{C}}_{\mathrm{s}\mathrm{x}}^{\mathrm{j}\mathrm{i}}& {\mathrm{C}}_{\mathrm{s}\mathrm{x}}^{\mathrm{j}\mathrm{j}}& 0& 0\\ 0& 0& {\mathrm{C}}_{\mathrm{s}\mathrm{y}}^{\mathrm{i}\mathrm{i}}& {\mathrm{C}}_{\mathrm{s}\mathrm{y}}^{\mathrm{i}\mathrm{j}}\\ 0& 0& {\mathrm{C}}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{i}}& {\mathrm{C}}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{j}}\end{array}\right];\left[\begin{array}{cccc}{\mathrm{K}}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{i}}& {\mathrm{K}}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{j}}& 0& 0\\ {\mathrm{K}}_{\mathrm{s}\mathrm{x}}^{\mathrm{j}\mathrm{i}}& {\mathrm{K}}_{\mathrm{s}\mathrm{x}}^{\mathrm{j}\mathrm{j}}& 0& 0\\ 0& 0& {\mathrm{K}}_{\mathrm{s}\mathrm{y}}^{\mathrm{i}\mathrm{i}}& {\mathrm{K}}_{\mathrm{s}\mathrm{y}}^{\mathrm{i}\mathrm{j}}\\ 0& 0& {\mathrm{K}}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{i}}& {\mathrm{K}}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{j}}\end{array}\right]; \\ {u}_{\mathrm{s}}=\left[\begin{array}{c}{\mathrm{x}}_{\mathrm{s}}^{\mathrm{i}}\\ {\mathrm{x}}_{\mathrm{s}}^{\mathrm{j}}\\ {\mathrm{y}}_{\mathrm{s}}^{\mathrm{i}}\\ {\mathrm{y}}_{\mathrm{s}}^{\mathrm{j}}\end{array}\right];P_{\mathrm{s}}=\left[\begin{array}{c}{\mathrm{F}}_{\mathrm{x}\mathrm{s}}^{\mathrm{i}}\\ {\mathrm{F}}_{\mathrm{x}\mathrm{o}}^{\mathrm{j}}+{\mathrm{F}}_{\mathrm{x}\mathrm{t}}^{\mathrm{j}}\\ {\mathrm{F}}_{\mathrm{y}\mathrm{s}}^{\mathrm{i}}\\ {\mathrm{F}}_{\mathrm{y}\mathrm{o}}^{\mathrm{j}}+{\mathrm{F}}_{\mathrm{y}\mathrm{t}}^{\mathrm{j}}\end{array}\right]. \end{gathered}$$

where ${\mathrm{F}}_{\mathrm{x}\mathrm{s}}^{\mathrm{i}},{\mathrm{F}}_{\mathrm{y}\mathrm{s}}^{\mathrm{i}}$ is the resultant force of radial electromagnetic force in x, y direction on the internal degree-of-freedom of stator structure;${\mathrm{F}}_{\mathrm{x}\mathrm{o}}^{\mathrm{j}},{\mathrm{F}}_{\mathrm{y}\mathrm{o}}^{\mathrm{j}}$ is the force in x, y direction on the interface between stator structure s and connecting substructure; and ${\mathrm{F}}_{\mathrm{x}\mathrm{t}}^{\mathrm{j}},{\mathrm{F}}_{\mathrm{y}\mathrm{t}}^{\mathrm{j}}$ is the constraint reaction force in x, y direction on the constraint surface of stator substructure.

The modal matrix of the stator can be obtained from the following formula [18]:

$$\left(K_{\mathrm{s}}-{\lambda }_{\mathrm{s}}{M}_{\mathrm{s}}\right){\mathsf{\omega }}_{\mathrm{s}}=O$$

(14)

Further, the modal matrix can be expressed as

$$u_{\mathrm{s}}={\mathsf{\omega }}_{\mathrm{s}}^{\mathrm{k}}{q}_{\mathrm{s}}=\left[\begin{array}{l}\begin{array}{cc}{\mathsf{\omega }}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{k}}& 0\\ {\mathsf{\omega }}_{\mathrm{s}x}^{\mathrm{j}\mathrm{k}}& 0\end{array}\\ \begin{array}{cc}0& {\mathsf{\omega }}_{\mathrm{s}\mathrm{y}}^{\mathrm{i}\mathrm{k}}\\ 0& {\mathsf{\omega }}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{k}}\end{array}\end{array}\right]\left[\begin{array}{l}{\mathrm{q}}_{\mathrm{s}\mathrm{x}}\\ {\mathrm{q}}_{\mathrm{s}\mathrm{y}}\end{array}\right]=\left[\begin{array}{c}{\mathsf{\omega }}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{k}}{\mathrm{q}}_{\mathrm{s}\mathrm{x}}\\ {\mathsf{\omega }}_{\mathrm{s}\mathrm{x}}^{\mathrm{j}\mathrm{k}}{\mathrm{q}}_{\mathrm{s}\mathrm{x}}\\ {\mathsf{\omega }}_{\mathrm{s}\mathrm{x}}^{\mathrm{i}\mathrm{k}}{\mathrm{q}}_{\mathrm{s}\mathrm{y}}\\ {\mathsf{\omega }}_{\mathrm{s}\mathrm{y}}^{\mathrm{j}\mathrm{k}}{\mathrm{q}}_{\mathrm{s}\mathrm{y}}\end{array}\right]$$

(15)

where q_s is the stator modal coordinate. Its vibration shape is shown in Figure 9.

As can be seen in Figure 9, the improved stator has a more stable structure, a better stability performance in the operation process and a very significant improvement in the vibration amplitude.

4. Analysis of Rotor Improvement

4.1. Introduction to Rotor Structure

Figure 10 shows two old rotor structures of a permanent magnet rotor deflection type 3-DOF motor, i.e., drum and butterfly shapes. The drum shaped rotor is part of the ball, and the outer surface has a spherical outline. The butterfly rotor is divided into three layers, which are alternately distributed in the circular direction according to the law of N and S poles. There is a certain gap between each layer of rotor, and there are different angles between the three layers and the magnet center. It has been proved that the use of a butterfly rotor can effectively improve the tilt angle and tilt torque of the motor and has better multi-DOF motion characteristics. The improved rotor inherits some features of the butterfly rotor and still uses the butterfly surface, but in the form of fragmentation on the permanent magnet pole in the axial direction, which can make the permanent magnet waveform stronger. In the axial direction, the same pole is divided into three parts. At the same time, a permanent magnet is added between the two opposite poles to connect the opposite poles in series. It has been proved that the air gap magnetic density amplitude can be increased by about 10% by using this kind of permanent magnet structure.

4.2. Magnetic Field of Rotor

The magnetic field produced by the rotor can be obtained from the Laplace equation [19]. The flux density of the air gap can be obtained by solving the Laplace equation in the spherical coordinate system. The magnetic field flux density of the air gap is expressed as

$$B_{air}=\left[\begin{array}{l}{B}_{airr}\\ B_{air\theta }\\ B_{air\varphi }\end{array}\right]=\left[\begin{array}{l}{\displaystyle \sum _{n=2,4,\dots }^{\infty }{\displaystyle \sum _{m=\pm 2,\pm 10,\dots }^{\pm n}\left(\left(n+1\right){\mu }_r{B}_n^m\frac{1}{r^{n+2}}\right)Y_n^m\left(\theta ,\varphi \right)}}\\ {\displaystyle \sum _{n=2,4,\dots }^{\infty }{\displaystyle \sum _{m=\pm 2,\pm 10,\dots }^{\pm n}\left(-{\mu }_r{B}_n^m\frac{1}{r^{n+2}}\right)\frac{\partial Y_n^m\left(\theta ,\varphi \right)}{\partial \theta }}}\\ {\displaystyle \sum _{n=2,4,\dots }^{\infty }{\displaystyle \sum _{m=\pm 2,\pm 10,\dots }^{\pm n}\left(-{\mu }_r{B}_n^m\frac{1}{r^{n+2}}\right)\frac{1}{\sin \theta }\frac{\partial Y_n^m\left(\theta ,\varphi \right))}{\partial \varphi }}}\end{array}\right]$$

(16)

where $Y_n^m(\theta ,\varphi )$ is the spherical harmonic function whose value is related to n and m. The coefficients $A_n^m$, $B_n^m$, m and n are undetermined, which is determined by the boundary conditions.

The distribution of air gap magnetic field flux density can be obtained by superposing the magnetic field flux density in each direction. Figure 11 shows the comparison of three components of magnetic field flux density. It can be seen from the figure that the improved stator for radial magnetic density can have a more stable sinusoidal law. Although there is a certain degree of attenuation on the amplitude, at the same time, the waveform becomes more stable, and there is a certain degree of weakening of the radial electromagnetic force. For radial magnetic field flux density, the result is positive. For non-radial magnetic field flux density, the improved stator structure can make the waveform smoother and increase the amplitude. For non-sinusoidal flux density, the result can also be positive.

Figure 12 shows the radial magnetic field flux density distribution features of several different rotor permanent magnet structures at the center section. It can be seen from the figure that B_r and B_φ at the center section of the motor with drum and butterfly rotors are basically the same, without much deviation. After adopting the improved rotor, B_r and B_φ at the center section of the motor have changed to a large extent, and the magnitude of B_r has been significantly increased. The amplitude of B_φ is restrained to a great extent; especially after the new rotor structure is adopted, B_r is further optimized and B_φ is further restrained. The impact on B_θ is too small, so B_θ is not considered, only B_r and B_θ are analyzed. This kind of optimization effect is expected to be obtained by the motor, so the improved rotor structure can play an obvious role in the optimization of the motor. The flux density of the motor before improvement can be obtained using the original magnetic field analysis method, and the flux density of the motor after improvement can be obtained using the following formula:

$$B=K_m\frac{B_{re}}{\pi \sigma }\arctan \frac{a_m{b}_m}{2\delta \sqrt{4{\delta }^2+a_m^2+b_m^2}},$$

(17)

where K_m is the end face coefficient of the short side length of the permanent magnet pole, B_re is the permanent magnet remanence, σ is the leakage coefficient, a_m is the short side length of the permanent magnet pole, b_m is the long side length of the permanent magnet pole, and δ is the air gap length.

The results of the series connection of the permanent magnet can be expressed as follows:

$$B=K_m\frac{1.1B_{re}}{\pi \sigma }\arctan \frac{a_m{b}_m}{2\delta \sqrt{4{\delta }^2+a_m^2+b_m^2}},$$

(18)

4.3. Analysis of Motor Torque

The calculation of torque needs to use the parameters involved in the stator coils. The rotation of the motor cannot be completed only by relying on the rotor. The motor can rotate because of the interaction of the magnetic field generated by the rotor pole and energization of the stator coils [20]. The parameter annotation of the stator coil is shown in Figure 13. Firstly, the single coil torque model is used, and the single coil torque model produces three directions of x, y and z. The electromagnetic torque can be expressed as

$$\{\begin{array}{l}{T}_z={\displaystyle \underset{R_1}{\overset{R_2}{\int }}{\displaystyle \underset{{\zeta }_1}{\overset{\zeta }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{r}^{-3}{B}_r\sin \left(\varphi \right)\sin \left(\theta \right)\sin \left(\zeta \right)}}}\mathrm{d}\varphi \mathrm{d}\zeta \mathrm{d}r\\ T_x={\displaystyle \underset{R_1}{\overset{R_2}{\int }}{\displaystyle \underset{{\zeta }_1}{\overset{\zeta }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{r}^{-3}{B}_r\left(\cos \varphi +{\cos }^2\varphi \cos \theta \right)\sin \zeta }}}\mathrm{d}\varphi \mathrm{d}\zeta \mathrm{d}r\\ T_y={\displaystyle \underset{R_1}{\overset{R_2}{\int }}{\displaystyle \underset{{\zeta }_1}{\overset{\zeta }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{r}^{-3}{B}_r\left(\cos \varphi -{\cos }^2\varphi \cos \theta \right)\sin \zeta }}}\mathrm{d}\varphi \mathrm{d}\zeta \mathrm{d}r\end{array},$$

(19)

where R₁ is the distance from the inner diameter of stator coil to the center of motor, and R₂ is the distance from the outer diameter of stator coil to the center of the motor.

Figure 14 shows the comparison of the finite element and analytical methods of different rotor rotation torque and deflection torque. Through Figure 14, it can be found that the error between the finite element method and analytical method is in a very limited range, which verifies the accuracy of the theory.

Figure 15 shows the torque curves of the analytical method corresponding to different rotors. The rotation torque is the comparison diagram of the torque of the central section of the motor and the comparison diagram of the torque when the deflection torque is θ = 60°. The improved rotor rotation torque and the deflection torque have been significantly improved. The amplitude and change rule of T_x and T_y are the same, but there is a phase difference in the mechanical angle. Here, it is just an example, which represents that the two pictures are exactly the same.

Table 1. Improving the main parameters of motor.

Parameter	Drum	Butterfly	Layered	Series	Optimization Rate
Bmax (T)	0.55	0.57	0.9	0.99	80%
Tzmax (B·m)	0.24	0.24	0.28	0.32	25%
Txmax (B·m)	0.2	0.205	0.25	0.28	40%
Vmag (mm3)	6021.39	5633.66	2342.01	2732.40	54.72%

describes the optimization results of the main parameters of the improved rotor. The optimization rate refers to the results obtained by comparing the parameters between the drum rotor and the series PM rotor. The optimization result o can be expressed as

$$O=\frac{P_d-P_s}{P_d}\times 100\%,$$

(20)

where P_d represents the parameters of drum rotor, and P_s represents the parameters of improved series rotor.

Table 2. Relevant parameters of motor.

Parameter	Value
Out diameter (mm)	150
Axial length (mm)	120
PM remanence (T)	1.15
PM relative permeability	1.05
Inner diameter (mm)	75

shows the relevant parameters of the motor for reference.

Figure 16 shows the experimental torque measurement equipment. The experiment is mainly composed of the prototype, sensor and stepping motor. The sensor shaft is connected to the installation side of the sensor. Since the sensor is of very high stiffness, the output shaft together with the sensor and the sensor shaft can be regarded as a whole rigid part. The arc guide track is used to restrict the sensor shaft and indicate the tilting angle. The slide block can be fixed by means of four screws so that the sensor shaft cannot tilt. Similarly, the sensor shaft itself can also be pressed by two brake blocks inside the slide block so that it cannot rotate. Thus, the rotor position and the number of degrees-of-freedom becomes controllable.

5. Conclusions

In this paper, an improved scheme for the structure of the 3-DOF motor with a permanent magnet deflection type rotor is proposed, and a simulated comparison is conducted and verified. The optimization of the structure can improve the performance of the motor. The improved stator structure can optimize the air gap flux density by 22.73%, and greatly enhance the stability of the structure. Using the improved rotor structure, a larger magnetic density amplitude can be obtained, and the optimization degree is as high as 80%. At the same time, it can achieve 25% of the optimization effect of the rotation torque, 40% of the optimization effect of the tilt torque and 54.72% of the material consumption of the permanent magnet. The positive effect of the improved structure is obvious and provides the reference for further design and optimization of relative multi-DOF motors or actuators.