Analysis of the air gap magnetic field in cylindrical magnetic couplings based on mathematical and finite element approach | Scientific Reports

Scientific Reports volume 14, Article number: 27067 (2024) Cite this article

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The air gap magnetic field of a magnetic coupling plays a crucial role in the examination of its static torque and eddy current losses. Precise characterization of the air gap magnetic field is essential for ensuring the accuracy of performance assessments of the magnetic coupling. This study focuses on the cylindrical magnetic coupling as the subject of investigation, employing mathematical analysis and finite element methods to evaluate and quantify the magnetic field properties. The study establishes a mathematical model of the magnetic field of a magnetic coupling based on electromagnetic field principles and the superposition theorem. An analytical formula for the magnetic flux density distribution of the air-gap magnetic field is derived. Subsequently, a three-dimensional magnetic field finite element model is created using the finite element method to numerically calculate the air gap magnetic field and validate the analytical formula. The research also explores the periodic distribution characteristics of the magnetic field in the air gap, analyzing axial differences in magnetic flux density value and direction distribution influenced by end effects.

Magnetic coupling, a non-contact torque transmission mechanism utilizing magnetic force, offers improved reliability and sealing performance in mechanical operations. It provides advantages such as shock absorption, smooth operation, overload protection, simple structure, easy maintenance, low noise, and long service life1,2. Current research on magnetic couplings primarily focuses on the air gap magnetic field, static torque, and eddy current loss of the isolation sleeve. Understanding the accurate distribution of the air gap magnetic field is crucial for analyzing the static torque and eddy current losses of the magnetic coupling. Eddy current losses can generate heat, raising the overall temperature of the magnetic coupling and potentially affecting the magnetism of permanent magnets, leading to demagnetization. It seriously affects the normal use of magnetic couplings. Therefore, it is extremely important to master the magnetic field characteristics of magnetic couplings.

Magnetic field characteristics are important parameters that affect the performance of magnetic couplings3,4,5. The distribution pattern of the magnetic field in the air gap of a magnetic coupling is one of the important factors determining the maximum torque of the magnetic rotor. The commonly research methods include formula induction method and finite element calculation method. The formula induction method is based on the electromagnetic induction principle, Maxwell equation, equivalent charge method, and other theoretical foundations6,7,8. Scholars analyzed the characteristics of different magnetic circuits in magnetic couplings and tested their performance under different magnetic circuit structures. They provided several empirical formulas for the engineering design of magnetic couplings, and based on these formulas, analyzed and designed the couplings9,10,11. Starting from the air gap magnetic field, the calculation formula for the driving torque of the magnetic couplings was derived, and the relationship between the air gap magnetic field distribution, transmitted force, and torque of the magnetic coupling was obtained. By analyzing the distribution of the air gap magnetic field, a relevant finite element model is established to calculate the eddy current loss value of the magnetic coupling12. By studying the magnetic field of synchronous motors and segmenting the magnetic blocks, structural optimization was carried out to reduce eddy current losses and energy losses13. The equivalent magnetic charge method was used to establish and calculate the air gap magnetic field intensity of the magnetic coupling, and the distribution pattern of magnetic induction intensity in the air gap area was obtained14. The distribution patterns of magnetic induction intensity and magnetic force magnitude under different states were analyzed and calculated using finite element method, and the design of relevant parameters was optimized15.

The study of the air gap magnetic field is the theoretical basis for analyzing the transmitted torque and eddy current losses16. Therefore, a simple and accurate analysis of the distribution and size of the air gap magnetic field is a guarantee for the reliability of the research on the transmission performance of magnetic couplings. This paper conducts research on the magnetic field characteristics of magnetic couplings through theoretical analysis and simulation, which is of great significance for improving the safety and reliability of magnetic couplings. Firstly, the structural parameters of the magnetic coupling are presented. Secondly, the air gap magnetic field of the magnetic coupling is analytically calculated, using theoretical analysis methods. Thirdly, the three-dimensional magnetic field finite element calculation model of the magnetic coupling is established, and the magnetic field characteristics are analyzed and calculated. Fourthly, the influence of end effect on magnetic field characteristics is analyzed.

The cylindrical magnetic coupling was used in this study, consisting of an inner rotor, an outer rotor, and an isolation sleeve, as shown in Fig. 1. The inner rotor diameter of the cylindrical magnetic coupling is slightly smaller, while the outer rotor diameter is slightly larger. The magnets were installed on the inner side of the outer rotor and the outer side of the inner rotor, both arranged in an even number, with adjacent magnets having opposite magnetization directions. The cylindrical magnetic coupling can increase the torque by increasing the coupling length of the inner and outer magnetic rotors.

The physical image of the magnetic coupling is shown in Fig. 2. In this study, the magnetic coupling were running at 900 rpm. The maximum pressure inside the isolation sleeve is 1.0 MPa.

Cylindrical Magnetic Coupling. (1-external rotor, 2-inner magnet, 3-isolation sleeve, 4-external magnet, 5-inner rotor).

The physical image of the magnetic coupling. (a- inner rotor, b-outer rotor).

In this study, ND-FE-B(N45) was chosen as the material for both the inner and external magnet due to its large remanence and high magnetic energy product. In order to improve circuit performance and reduce magnetic leakage, Q235A was chosen for both the inner and external rotor. The isolation sleeve was made from TC4 titanium alloy, known for its exceptional performance. Specific size parameters and materials for the magnetic coupling are shown in Table 1.

The gap between the inner rotor and outer rotor of the magnetic coupling is called the air gap. Its size has a significant impact on the working performance of magnetic couplings, especially on torque. The smaller the air gap, the less magnetic flux is consumed between the air gaps, and the greater the torque. However, considering the relative motion between the inner and outer magnetic rotors and the isolation sleeve, it is necessary to leave appropriate motion space. If the air gap is too small, friction between moving parts is prone to occur. Usually, the air gap size of magnetic couplings is more suitable at 2–10 mm.

In the analytical calculation, the air gap magnetic field of the magnetic coupling is assumed as follows:

1) Permanent magnets have linear demagnetization characteristics, with uniform magnetization along the radial direction, and their residual magnetization vector is\(M={{{B_r}} \mathord{\left/ {\vphantom {{{B_r}} {{\mu _0}}}} \right. \kern-0pt} {{\mu _0}}}\), \({B_r}\) is the residual magnetic flux density, \({\mu _0}\) is the vacuum permeability.

2) The size of the air gap is much smaller than the axial height, ignoring the end effect.

3) Neglecting the effect of eddy current on the air gap magnetic field.

4) the rotor of the magnetic coupling is thick enough to prevent magnetic saturation.

The analytical calculation of the air gap magnetic field of the magnetic coupling mainly uses the Gaussian magnetic law \(\nabla B=0\) in the Maxwell equation system and constitutive relationship\(B=\mu H\), where H - magnetic field strength, A/m; B - magnetic induction intensity, T; \(\mu\)- the magnetic permeability of the medium, H/m.

As shown in Fig. 3, the air gap magnetic field of the magnetic coupling is composed of tile shaped magnets distributed on the inner and outer rotors, which can be regarded as the superposition of the air gap magnetic field formed by the outer rotor alone and the air gap magnetic field formed by the inner rotor alone. For the air gap magnetic field of a single rotor, its composition is similar to that of a surface mounted brushless permanent magnet motor. Based on the analysis method of the air gap magnetic field of a permanent magnet motor17,18,19,20,21, some adaptive improvements can be made to the analytical formula according to the structural characteristics of the magnetic rotor itself. Therefore, the analytical formula for the air gap magnetic field of the permanent magnet coupling is obtained.

Air gap magnetic field model of magnetic coupling.

where p is the number of magnetic poles of the outer or inner magnet, p = 12; µ0 is the vacuum magnetic permeability; n is the rotational speed, rpm.

According to the Fourier expansion formula, the residual magnetization of the radial and tangential components of the magnetic steel is expressed as.

\({M_r}=\sum\limits_{{n=1,3,5 \cdots }}^{\infty } {{M_{rn}}\cos \left( {np\theta } \right)}\)

\({M_\theta }=0\)

\({M_{rn}}=\frac{{2{B_r}}}{{{\mu _0}}}\frac{{\sin \frac{{n\pi }}{2}}}{{\frac{{n\pi }}{2}}}\)

The scalar magnetic potential distribution of the air gap magnetic field is applicable to the Laplace’s equation, expressed as.

\(\frac{{{\partial ^2}{\varphi _{air}}}}{{\partial {r^2}}}+\frac{1}{r}\frac{{\partial {\varphi _{air}}}}{{\partial r}}+\frac{1}{{{r^2}}}\frac{{{\partial ^2}{\varphi _{air}}}}{{\partial {\theta ^2}}}=0\)

The magnetic field is applicable to the quasi Poisson equation, which is expressed as.

\(\frac{{{\partial ^2}{\varphi _{magnet}}}}{{\partial {r^2}}}+\frac{1}{r}\frac{{\partial {\varphi _{magnet}}}}{{\partial r}}+\frac{1}{{{r^2}}}\frac{{{\partial ^2}{\varphi _{magnet}}}}{{\partial {\theta ^2}}}=\frac{1}{{{\mu _r}}}div\overrightarrow M\)

The relationship between scalar magnetic potential and magnetic field strength H and magnetic induction strength B is expressed as.

\({H_r}= - \frac{{\partial \varphi }}{{\partial r}}\)

\({H_\theta }= - \frac{1}{r}\frac{{\partial \varphi }}{{\partial \theta }}\)

\(\overrightarrow B = - {\mu _0}\nabla \overrightarrow \varphi\)

The boundary conditions for the air gap magnetic field formed by the inner magnet are as follows:

\(\left\{ \begin{gathered} {H_{\theta air}}\left( {r,\theta } \right)\left| {_{{r={R_4}}}=0} \right. \hfill \\ {H_{\theta magnet}}\left( {r,\theta } \right)\left| {_{{r={R_1}}}=0} \right. \hfill \\ {H_{\theta air}}\left( {r,\theta } \right)\left| {_{{r={R_2}}}={H_{\theta magnet}}\left( {r,\theta } \right)\left| {_{{r={R_2}}}} \right.} \right. \hfill \\ {B_{\theta air}}\left( {r,\theta } \right)\left| {_{{r={R_2}}}={B_{\theta magnet}}\left( {r,\theta } \right)\left| {_{{r={R_2}}}} \right.} \right. \hfill \\ \end{gathered} \right.\)

The boundary conditions for the air gap magnetic field formed by the external magnet are as follows:

\(\left\{ \begin{gathered} {H_{\theta air}}\left( {r,\theta } \right)\left| {_{{r={R_1}}}=0} \right. \hfill \\ {H_{\theta magnet}}\left( {r,\theta } \right)\left| {_{{r={R_4}}}=0} \right. \hfill \\ {H_{\theta air}}\left( {r,\theta } \right)\left| {_{{r={R_3}}}={H_{\theta magnet}}\left( {r,\theta } \right)\left| {_{{r={R_3}}}} \right.} \right. \hfill \\ {B_{\theta air}}\left( {r,\theta } \right)\left| {_{{r={R_3}}}={B_{\theta magnet}}\left( {r,\theta } \right)\left| {_{{r={R_3}}}} \right.} \right. \hfill \\ \end{gathered} \right.\)

The Laplace’s equation and quasi Poisson equation are solved according to the boundary conditions. The radial magnetic flux density and tangential magnetic flux density in the air gap magnetic field are expressed as follows:

\({B_{rair}}\left( {r,\theta } \right)=\sum\limits_{{n=1,3,5 \cdots }}^{\infty } {{K_B}} \left( n \right){f_{{B_r}}}\left( r \right) \cdot \cos \left( {np\theta } \right)\)

\({B_{\theta air}}\left( {r,\theta } \right)=\sum\limits_{{n=1,3,5 \cdots }}^{\infty } {{K_B}} \left( n \right){f_{{B_\theta }}}\left( r \right) \cdot \sin \left( {np\theta } \right)\)

The coefficient of the air gap magnetic field formed solely by the inner magnetic rotor is expressed as:

\({K_B}\left( n \right)={K_{in}}\left( n \right)=\frac{{{\mu _0}{M_{rn}}}}{{{\mu _r}}}\frac{{np}}{{{{\left( {np} \right)}^2} - 1}}\left\{ {\frac{{\left( {np - 1} \right)+2{{\left( {\frac{{{R_1}}}{{{R_2}}}} \right)}^{np+1}} - \left( {np+1} \right){{\left( {\frac{{{R_1}}}{{{R_2}}}} \right)}^{2np}}}}{{\frac{{{\mu _r}+1}}{{{\mu _r}}}\left[ {1 - {{\left( {\frac{{{R_1}}}{{{R_4}}}} \right)}^{2np}}} \right] - \frac{{{\mu _r} - 1}}{{{\mu _r}}}\left[ {{{\left( {\frac{{{R_2}}}{{{R_4}}}} \right)}^{2np}} - {{\left( {\frac{{{R_1}}}{{{R_2}}}} \right)}^{2np}}} \right]}}} \right\}\)

\({f_{{B_r}}}\left( r \right)={f_{{B_r}in}}\left( r \right)={\left( {\frac{r}{{{R_4}}}} \right)^{np - 1}}{\left( {\frac{{{R_2}}}{{{R_4}}}} \right)^{np+1}}+{\left( {\frac{{{R_2}}}{r}} \right)^{np+1}}\)

\({f_{{B_\theta }}}\left( r \right)={f_{{B_\theta }in}}\left( r \right)= - {\left( {\frac{r}{{{R_4}}}} \right)^{np - 1}}{\left( {\frac{{{R_2}}}{{{R_4}}}} \right)^{np+1}}+{\left( {\frac{{{R_2}}}{r}} \right)^{np+1}}\)

If there is an angle difference Ф between the inner and outer rotors, the coefficient of the air gap magnetic field formed by the internal magnet alone is expressed as.

\({B_{rair,in}}\left( {r,\theta } \right)=\sum\limits_{{n=1,3,5 \cdots }}^{\infty } {{K_{in}}} \left( n \right){f_{{B_r}in}}\left( r \right) \cdot \cos \left[ {np\left( {\theta +\phi } \right)} \right]\)

\({B_{\theta air,in}}\left( {r,\theta } \right)=\sum\limits_{{n=1,3,5 \cdots }}^{\infty } {{K_{in}}} \left( n \right){f_{{B_\theta }in}}\left( r \right) \cdot \sin \left[ {np\left( {\theta +\phi } \right)} \right]\)

The coefficient of the air gap magnetic field formed solely by the external magnetic rotor is expressed as:

\({K_B}\left( n \right)={K_{ex}}\left( n \right)=\frac{{ - {\mu _0}{M_{rn}}}}{{{\mu _r}}}\frac{{np}}{{{{\left( {np} \right)}^2} - 1}}\left\{ {\frac{{\left( {np - 1} \right){{\left( {\frac{{{R_3}}}{{{R_4}}}} \right)}^{2np}}+2{{\left( {\frac{{{R_3}}}{{{R_4}}}} \right)}^{np - 1}} - \left( {np+1} \right)}}{{\frac{{{\mu _r}+1}}{{{\mu _r}}}\left[ {1 - {{\left( {\frac{{{R_1}}}{{{R_4}}}} \right)}^{2np}}} \right] - \frac{{{\mu _r} - 1}}{{{\mu _r}}}\left[ {{{\left( {\frac{{{R_1}}}{{{R_3}}}} \right)}^{2np}} - {{\left( {\frac{{{R_3}}}{{{R_4}}}} \right)}^{2np}}} \right]}}} \right\}\)

\({f_{{B_r}}}\left( r \right)={f_{{B_r}ex}}\left( r \right)={\left( {\frac{r}{{{R_3}}}} \right)^{np - 1}}+{\left( {\frac{{{R_1}}}{{{R_3}}}} \right)^{np - 1}}{\left( {\frac{{{R_1}}}{r}} \right)^{np+1}}\)

\({f_{{B_\theta }}}\left( r \right)={f_{{B_\theta }ex}}\left( r \right)= - {\left( {\frac{r}{{{R_3}}}} \right)^{np - 1}}+{\left( {\frac{{{R_1}}}{{{R_3}}}} \right)^{np - 1}}{\left( {\frac{{{R_1}}}{r}} \right)^{np+1}}\)

By superposing the air gap magnetic field formed by the inner magnetic rotor and the outer magnetic rotor, and the radial and tangential magnetic flux densities of the air gap magnetic field can be expressed as.

\({B_{rair}}\left( {r,\theta } \right)=\sum\limits_{{n=1,3,5 \cdots }}^{\infty } {\left\{ {{K_{in}}\left( n \right){f_{{B_r}in}}\left( r \right) \cdot \cos \left[ {np\left( {\theta +\phi } \right)} \right]+{K_{ex}}\left( n \right){f_{{B_r}ex}}\left( r \right) \cdot \cos \left( {np\theta } \right)} \right\}}\)

\({B_{\theta air}}\left( {r,\theta } \right)=\sum\limits_{{n=1,3,5 \cdots }}^{\infty } {\left\{ {{K_{in}}\left( n \right){f_{{B_\theta }in}}\left( r \right) \cdot \sin \left[ {np\left( {\theta +\phi } \right)} \right]+{K_{ex}}\left( n \right){f_{{B_\theta }ex}}\left( r \right) \cdot \sin \left( {np\theta } \right)} \right\}}\)

In summary, the magnetic flux density of the air gap magnetic field of a magnetic coupling can be expressed as.

\(B\left( {r,\theta } \right)=\sqrt {{B_{rair}}{{\left( {r,\theta } \right)}^2}+{B_{\theta air}}{{\left( {r,\theta } \right)}^2}}\)

Based on the structural parameters of the magnetic coupling given in the previous section, the circumferential variation trends of the radial magnetic flux density, tangential magnetic flux density, and total magnetic flux density values of the air gap magnetic field at r = 56.5 mm can be calculated when the magnetic coupling is under rated operating conditions, as shown in Figs. 4, 5 and 6.

It can be seen that the radial magnetic flux density, tangential magnetic flux density, and total magnetic flux density values in the air gap magnetic field all exhibit a periodic distribution along the circumference. The radial magnetic flux density curve is approximately sinusoidal in distribution, with a maximum value of 0.78T. And there are six cycles of variation throughout the entire circumference. The value of tangential magnetic flux density is relatively small, with a maximum value of 0.34T. The total magnetic flux density exists for twelve cycles within a circle, with a maximum value of 0.8T.

Radial magnetic flux density of air gap magnetic field.

Tangential magnetic flux density of air gap magnetic field.

Magnetic flux density of air gap magnetic field.

The three-dimensional finite element model of the magnetic coupling is shown as Fig. 7. The inner rotor and outer rotor are moving components, thus forming two moving domains. Two polyhedral cylinders are used to wrap the inner rotor and outer rotor respectively, forming two cylindrical rotation domains. The boundary of the motion domain is located halfway through the air gap and assigned a corresponding rotational speed.

The three-dimensional finite element model of the magnetic coupling (1-external rotor, 2-inner rotor, 3-isolation sleeve, 4-rotational domain, 5-external magnet, 6-inner magnet, 7-computational domain).

When the magnetic coupling operates under rated conditions, the distribution of magnetic flux density on the inner and outer rotors is shown in Fig. 8. For the inner and outer magnetic rotors, the maximum magnetic flux density is 2.66T, which occurs at the junction of the magnet and the magnetic conductor. This phenomenon is called the sharp angle effect22, which the magnetic field lines gather at the sharp corners of the junction between the N, S poles of the magnet and the magnetic conductor. The distribution characteristics of this magnetic flux density fully reflect the role of the magnetic conductor in the magnetic coupling, which controls the direction of the magnetic induction line, changes the magnetic flux density, reduces magnetic leakage, and shields the interference of external magnetic fields.

Magnetic flux density distribution on inner and outer rotors.

Figure 9 shows the distribution of magnetic flux density on the isolation sleeve. It can be seen that the magnetic flux density on the isolation sleeve is mainly distributed in the corresponding area of the magnet, with a maximum value of 0.74T. Affected by the arrangement of magnetic poles, the magnetic flux density on the isolation sleeve exhibits a regular periodic distribution in the circumferential direction, consisting of twelve cycles in total. Within each cycle, there is a staggered distribution of areas with higher magnetic flux density and areas with lower magnetic flux density. Due to the existence of angle difference, there are two radial alignment methods for the inner and outer magnets, namely the same pole relative and the opposite pole relative. At the same polarity, there are very few magnetic field lines passing vertically through the isolation sleeve due to the deviation of the magnetic field lines. The magnetic flux density at the isolation sleeve is very small, and the minimum point is close to zero. When the poles are opposite, the magnetic field lines starting from the N pole can directly pass through the isolation sleeve and return to the S pole of the opposite magnet. There are more magnetic field lines passing through the isolation sleeve, resulting in a higher magnetic flux density at the corresponding isolation sleeve. When the magnetic coupling is working, the inner and outer rotors are in a synchronous rotation state, and the magnetic field where the isolation sleeve is located constantly changes. The magnetic flux density on the isolation sleeve also changes, but the overall distribution characteristics on the isolation sleeve remain unchanged at different times.

Magnetic flux density distribution on the isolation sleeve.

The air gap magnetic field characteristics of magnetic couplings differ between the end and middle parts, known as the end effect. In order to simplify calculations, the end effect is often ignored, and it is believed that the distribution of the air gap magnetic field characteristics of magnetic couplings in the axial direction is the same, which can be equivalent to the air gap magnetic field characteristics of a certain cross-section. In fact, there are differences in the magnetic field characteristics of the air gap at different positions along the axial direction. The isolation sleeve is located in the air gap between the inner and outer magnets, and the magnetic flux density distribution on the sleeve is also affected by the end effect, thus affecting the distribution of eddy current losses23. The following will study the influence of end effects in the air gap magnetic field from both circumferential and axial directions.

The axial length of the magnet for the magnetic coupling selected in this study is 25 mm. A circle with a radius of 56.5 mm is drawn in the air gap corresponding to the magnet length of 12.5 mm and 25 mm, respectively, as the research path for the middle and end air gaps of the magnetic coupling. The comparison of the magnetic flux density values on these two paths is shown in Fig. 10.

It can be seen that the magnetic flux density curves of the end and middle air gaps are distributed periodically along the circumference, with twelve cycles within a circle, and the magnetic flux density value curve for each cycle is approximately half a sine. The peak magnetic flux density of the middle air gap is about twice that of the end air gap, and there is a certain deviation between the phase angles of the peaks. The maximum value of the middle air gap magnetic flux density is 0.74T, and the minimum value is 0.05T. The maximum value of the end air gap magnetic flux density is 0.36T, and the minimum value is 0.2T. This is due to the influence of the magnetic coupling structure and the relative position of the magnet, resulting in fewer magnetic field lines passing through the end air gap, resulting in a smaller end magnetic flux density value. The magnetic field lines at the middle air gap are constrained by the rotors and magnet. Most of the magnetic field lines develop radially within the air gap, while the magnetic field lines at the end position are less constrained by the rotors and magnet. The magnetic field lines disperse and develop in multiple directions in the air gap at the end, resulting in smaller differences in the magnetic flux density values of the air gap at the end, and the phase angle corresponding to the magnetic flux density peak is also different from that of the middle air gap.

Comparison of the magnetic flux density distribution along the circular path of the air gap magnetic field in the middle and end regions.

The magnetic poles on both sides of the air gap of the magnetic coupling can be isotropic and anisotropic. The magnetic flux density at the air gap of the opposite magnetic pole is relatively high, while the magnetic flux density at the air gap of the same magnetic pole is relatively low. Take two points a and b at the opposite magnetic pole and the same magnetic pole within the air gap with a radius of 56.5 mm. The paths of two points moving along the axis are denoted as lines_ A and line_ b. The magnetic flux density distribution along the axial direction of two points A and b is shown in Fig. 11. It can be clearly seen that the magnetic flux density is greatly affected by the end effect along the axis. The magnetic flux density of Line_ A and line_b varies symmetrically around the center at z = 12.5 mm, and the closer it is to the end, the greater the magnitude of the change in magnetic flux density. For line_ a, the end effect causes the magnetic flux density at the end to be smaller than that in the middle, while for line_ b, the end effect results in a higher magnetic flux density at the end compared to the middle part. Overall, the magnetic flux density difference between points a and b within the same axial length gradually increases from the end to the middle, which is consistent with the characteristics shown in Fig. 8.

Comparison chart of axial distribution of magnetic flux density.

Using the two circumferential paths at the air gap corresponding to the positions of 12. 5 mm and 25 mm in the length of the magnet in the previous section as the research path for the middle and end air gaps, the radial, tangential, and axial magnetic flux density is calculated, as shown in Figs. 12 and 13. Br is the radial magnetic flux density, Bphi is the tangential magnetic flux density, and Bz is the axial magnetic flux density. It can be seen that the components of the air gap magnetic flux density in three directions also exhibit a periodic distribution, with six cycles in a circular path24. For the middle air gap, the axial magnetic flux density is close to zero, the tangential component magnetic flux density is small, and the radial magnetic flux density distribution is approximately sinusoidal, with a maximum value of 0.72T and a minimum value of 0T. The distribution of magnetic flux density in the end air gap is consistent with that in the middle air gap. The radial magnetic flux density is approximately sinusoidal in distribution. The tangential and radial magnetic flux density values of the end air gap show a significant decrease compared to the middle air gap. For the radial magnetic flux density, the maximum value decreases to 0.30T, while the axial magnetic flux density of the end air gap shows a significant increase compared to the middle air gap, increasing from near zero in the middle air gap to 0.25T.

Directional distribution of magnetic flux density in the middle air gap.

Directional distribution of magnetic flux density in the end air gap.

In summary, due to the influence of end effects, the distribution direction of air gap magnetic flux density has also changed. The magnetic flux density in the middle air gap is mainly distributed radially, and the end air gap’s is mainly distributed axially. Comparing the results of the three-dimensional finite element calculation with two-dimensional analytical calculation, the magnetic flux density in the middle air gap is the closest in numerical and distribution directions, indicating that the analytical calculation results have a certain degree of reliability. The end effect has a significant impact on the magnetic flux density of the end gap, and the two-dimensional analytical calculation did not consider the end effect. Therefore, for two-dimensional analysis that ignores end effects, it is suitable for regular summary analysis and rough estimation. For calculating specific values, a three-dimensional calculation method that considers end effects should be used to be more accurate.

The eddy current losses are derived based on Faraday’s law of electromagnetic induction from Maxwell’s equations, expressed as \(\nabla E= - {{\partial B} \mathord{\left/ {\vphantom {{\partial B} {\partial t}}} \right. \kern-0pt} {\partial t}}\). Its physical meaning is that the curl of the electric field strength \(E\) is equal to the negative value of the change rate of the magnetic flux density \(B\) at that point over time. In addition, the constitutive relationship \(J=\sigma E\) must also be satisfied.

The curl of electric field strength in cylindrical coordinate system is expressed as:

\(\nabla \times \overrightarrow E =\left( {\frac{1}{r}\frac{{\partial {E_z}}}{{\partial \theta }} - \frac{{\partial {E_\theta }}}{{\partial z}}} \right)\overrightarrow {{e_r}} +\left( {\frac{{\partial {E_r}}}{{\partial z}} - \frac{{\partial {E_z}}}{{\partial r}}} \right)\overrightarrow {{e_\theta }} +\left( {\frac{{\partial {E_\theta }}}{{\partial r}} - \frac{1}{r}\frac{{\partial {E_r}}}{{\partial \theta }}} \right)\overrightarrow {{e_z}}\)

The magnetic flux density in cylindrical coordinate system is expressed as:

\(- \frac{{\partial \overrightarrow B }}{{\partial t}}= - \frac{{\partial \left( {{B_r}\overrightarrow {{e_r}} +{B_\theta }\overrightarrow {{e_\theta }} +{B_z}\overrightarrow {{e_z}} } \right)}}{{\partial t}}\)

The boundary conditions are expressed as:

\(\overrightarrow B ={B_r}\overrightarrow {{e_r}}\) ,\({B_\theta }=0\) ,\({B_z}=0\)

\(\overrightarrow E ={E_z}\overrightarrow {{e_z}}\), \({E_\theta }=0\), \({E_r}=0\), \(\frac{{\partial \overrightarrow E }}{{\partial r}}=0\)

By introducing boundary conditions, Faraday’s law of electromagnetic induction can be simplified as:

\(\frac{1}{r}\frac{{\partial {E_z}}}{{\partial \theta }}= - \frac{{\partial {B_r}}}{{\partial t}}\)

\(\frac{{\partial {E_z}}}{{\partial \theta }}= - {B_0}rm\cos \left( {m\theta } \right)\frac{{\partial \theta }}{{\partial t}}\)

\(\omega =\frac{{\partial \theta }}{{\partial t}}=\frac{{2\pi n}}{{60}}\)

\(\frac{{\partial {E_z}}}{{\partial \theta }}= - \frac{{2\pi nr{B_0}}}{{60}}\cos \left( {m\theta } \right)\)

By integration, it can be obtained as:

\(\int_{0}^{{E\left( \theta \right)}} {d{E_z}} = - \frac{{\pi r{B_0}}}{{30}}\int_{0}^{\theta } {\cos \left( {m\theta } \right)} d\left( {m\theta } \right)\)

\({E_z}\left( \theta \right)= - \frac{{\pi nrm{B_0}}}{{30}}\sin \left( {m\theta } \right)\)

According to the constitutive relationship between electric field strength and current density, the expression for current density is obtained as follows:

\(J=\sigma {E_z}\left( \theta \right)= - \frac{{\pi r\sigma nm{B_0}}}{{30}}\sin \left( {m\theta } \right)\)

The induced current in the unit area \(dA=rtd\theta\) is expressed as:

\(I=\frac{{\pi {r^2}\sigma nmt{B_0}}}{{30}}\sin \left( {m\theta } \right)d\theta\)

The eddy current loss in the unit area is expressed as:

\(dP={I^2}dR={\left[ {\frac{{\pi {r^2}\sigma nmt{B_0}^{2}}}{{30}}\sin \left( {m\theta } \right)d\theta } \right]^2}\frac{{dL}}{{\sigma rtd\theta }}\)

For an isolation sleeve with a magnetization length of \(L\), its eddy current loss is expressed as:

\(P=\int_{0}^{L} {\int_{0}^{{2\pi }} {\frac{{{\pi ^2}{r^2}\sigma {n^2}{m^2}t{B_0}^{2}}}{{900}}{{\sin }^2}\left( {m\theta } \right)d\theta } } dL=\frac{{{\pi ^3}{r^3}{n^2}B_{0}^{2}Lt\sigma }}{{900}}\)

Where \({B_0}\)is maximum magnetic flux density, T; \(L\) is magnetization length of isolation sleeve, mm;

\(t\) is wall thickness of isolation sleeve, mm; \(\sigma\) is conductivity of the isolation sleeve material, S/m.

It can be seen that the eddy current loss on the isolation sleeve is mainly affected by the radius, speed, magnetic induction intensity, magnetization length, thickness, and conductivity of the isolation sleeve material.

Figure 14 shows the distribution of eddy current loss density on the isolation sleeve under rated operating conditions. Eddy current loss is generated by induced current, so it is primarily concentrated in the area where the induced current is most prominent, namely, the projection position of the magnet on the isolation sleeve. The distribution characteristics of instantaneous magnetic flux density, induced current density and eddy current loss on the isolation sleeve are similar. At any given moment, there are 12 concentrated distribution areas of eddy current losses on the isolation sleeve, with the highest eddy current loss density corresponding to the area with the highest induced current density and magnetic flux density. The maximum eddy current loss density on the isolation sleeve is 4.25 × 106W/m3. Due to the continuous rotation of the inner and outer magnetic rotors during operation of the magnetic coupling, there is also rotational transformation in their respective magnetic fields. Therefore, at different times, the distribution of eddy current losses on the isolation sleeve also moves circumferentially with the rotation of the inner and outer magnets. Under the rated working state, the eddy current loss calculation result on the isolation sleeve as 29.2 W.

Cloud map of instantaneous eddy current loss density.

Based on the analytical method of the air gap magnetic field of permanent magnet motors, a mathematical model of the air gap magnetic field of magnetic couplings has been established. By treating the air gap magnetic field of the magnetic coupling as the superposition of the air gap magnetic field formed separately by the inner and outer rotors, an analytical formula is obtained. A three-dimensional finite element calculation model was established for the magnetic coupling, and the magnetic field characteristics were analyzed and calculated. The results show that the analytical calculation results are closest to the results of the middle air gap magnetic field calculated by the three-dimensional finite element method, indicating that the analytical calculation is reasonable and accurate. Affected by the arrangement of magnets, the magnetic flux density of the air gap magnetic field of the magnetic coupling exhibits a periodic distribution. The magnetic flux density of the middle air gap is mainly distributed radially. The closer it is to the end, the smaller the development direction of the magnetic induction line is constrained by the magnet and magnetic conductor, and the more divergent the distribution direction of the air gap magnetic flux density. Affected by the end effect, there are certain differences in the magnetic flux density values and distribution patterns of the air gap magnetic field in different planes along the length direction of the magnet.

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Funded by Open Foundation of Industrial Perception and Intelligent Manufacturing Equipment Engineering Research Center of Jiangsu Province (No. ZK21-05-08), Scientific Research Foundation of Nanjing Vocational University of Industry Technology (No. YK20-01-11), China Postdoctoral Science Foundation (Certificate Number: 2023M731674 ) and Jiangsu Funding Program for Excellent Postdoctoral Talent.

Industrial Perception and Intelligent Manufacturing Equipment Engineering Research Center of Jiangsu Province, Nanjing Vocational University of Industry Technology, Nanjing, 210023, PR China

Lian-bo Li

School of Energy and Power Engineering, Nanjing Institute of Technology, Nanjing, 211167, PR China

Tao Chen

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, PR China

Wei-xuan Li

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Lianbo Li wrote themain manuscript text. Tao Chen prepared data.Weixuan Li prepared figures and tables. All authors reviewed the manuscript.

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Li, Lb., Chen, T. & Li, Wx. Analysis of the air gap magnetic field in cylindrical magnetic couplings based on mathematical and finite element approach. Sci Rep 14, 27067 (2024). https://doi.org/10.1038/s41598-024-78673-z

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Received: 04 July 2024

Accepted: 04 November 2024

Published: 07 November 2024

DOI: https://doi.org/10.1038/s41598-024-78673-z

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