Special behaviors of two interacting permanent magnets with large different sizes

Abstract

Magnets are widely used in many fields, both industry and our daily life, because of their unique properties such as non-contact interactions and electricity generation. However, behaviors of permanent magnets seem to deviate from our basic understanding that opposite poles attract and like poles repel. We have recently discovered an interesting phenomenon in pairs of permanent magnets with large different sizes where the like poles do not seem to repel and the unlike poles do not seem to attract with each other, due to a localized area with reversed pole existing in the large permanent magnet. In this paper, we first experimentally confirm the existence of this particular phenomenon by studying interactions of several groups of permanent magnets with different sizes, then we introduce the property of the irreversible movements of load line and working point of permanent magnet to figure out this transition process from repulsion to attraction between a pair of permanent magnets with large different sizes. A reasonable explanation is finally given. In the end, we suggest a few applications for this special phenomenon.

Keywords

Magnetic force calculation load line demagnetizing field working point

1. Introduction

Magnets are used in many applications from simple toys to high-performance electromechanical systems, even some frontier researches such as vibration energy harvesters [1–5]. For permanent magnets, we readily accept what is already written in school textbooks: permanent magnets have two poles - north and south -, where like poles repel and unlike poles attract. For a pair of permanent magnets with same sizes arranged whose pole faces are faced to each other, this common sense is undoubted considering the situation that the middle magnetic axes of magnets coincide. We don’t pointedly discuss the situation that magnetic axes are far distant in this paper.

However, in more general situations, interacting permanent magnets do not have same sizes. Recently, in our study on the behaviors of interacting permanent magnets with large different sizes, we discover a special phenomenon that the localized area in the large magnet has a reversed pole, so it seems that not only repellence exists between like poles for a pair of permanent magnets but attraction; vice versa, not only attraction exists between unlike poles of permanent magnet pair but repellence.

In this paper, we describe the special phenomenon and verify it by repeating the experiment several times with different permanent magnets. First of all, we try to calculate the interacting magnetic force based on equivalent magnetizing current model to reveal the special phenomenon [6]. However, through the comparison between simulation and experiment, we realize that the present model is too limited to explain this phenomenon. Then, in order to observe the change of magnetic states, beginning with newly purchased magnets, step by step we measured the surface magnetic induction intensity of permanent magnets before and after every procedure of interactions. At last, we propose our analyses and explanation that irreversible demagnetizing process leads to shifts of load line and working point on demagnetizing curve.

2. Experiment

2.1. Measurement of interacting magnetic force

According to our previous study, the equivalent magnetizing current model could totally describe the interacting force between a pair of same-sized permanent magnets [5]. We design an experiment to measure the actual position-dependent magnetic force between two permanent magnets, as shown in Fig. 1. Here, we take an example of a pair of 10 mm × 10 mm × 2 mm Nd₂Fe₁₄B cuboid magnets and a pair of 𝜙6 mm × 2 mm Nd₂Fe₁₄B cylindrical magnets. One magnet of a pair is attached to the adjusted platform and the other one to the dynamometer (HF-5). The like magnetic poles are arranged face to face, it means the repulsive axial magnetic force exists between the magnet pairs in this experiment. We adjust the screws to simulate different relative positions of the magnets, while recording the displacements and forces by the laser displacement sensor (LK-G5001V) and dynamometer, respectively, whose minimum resolutions are 0.001 mm and 0.001 N.

Fig. 1.

Magnetic force measurement system. (a) cylindrical magnet pair, (b) cuboid magnet pair.

Fig. 2.

Schematic of the magnets moving state in the experiment, the 3D yellow arrows denote moving directions of magnets. (a) cylindrical magnets; (b) cuboid magnets.

In order to test the accuracy of the magnetic force calculation by using the equivalent magnetizing current model [5,6], we simulate the axial forces in z-axis when one magnet is moving over another that is fixed along x-axis. Here, we define the displacement of the moving magnet as zero when the projections of the pair of magnets coincide, as shown in Fig. 2, where magnet B is fixed and magnet A is movable. Figure 3 compares experimental measurements with the simulated axial magnetic forces F _Z as a function of the displacement x.

Fig. 3.

Comparison between simulation results and experimental measurements for (a) cylindrical magnets; (b) cuboid magnets.

It is observed that the simulation results are nicely consistent with the experimental measurement data. The accuracy of equivalent magnetizing current model is provisionally verified.

The above study is the interacting magnetic force calculation between magnets of the same size. Now if the interacting magnets’ sizes are of large difference, then the characters of the magnetic force cannot be generalized and the behaviors of the magnets cannot be considered as the same either.

Consider a block of N35 𝜙40 mm × 3 mm Nd₂Fe₁₄B cylindrical magnet and a block of N35 𝜙6 mm × 3 mm Nd₂Fe₁₄B cylindrical magnet with their like poles facing each other in this next study. With the same analytical method introduced above, we first calculate the interacting axial magnetic force F _Z between the magnets. The solid line in Fig. 4(a) shows the simulation results. Because the magnetizing currents exist around the outermost surface of the magnet, the largest repulsion mainly resides along the edges of the larger magnet, and the magnetic field is weaker in the middle [7,8]. This phenomenon also applies to the cuboid N35 Nd₂Fe₁₄B magnet pair whose sizes are 40 mm × 40 mm × 3 mm and 6 mm × 6 mm × 3 mm respectively, as the solid line shown in Fig. 4(b).

Fig. 4.

The simulation results of axial magnetic force interacting between (a) cuboid magnets and (b) cylindrical magnets of different sizes with comparisons to experimental measurements.

An interesting discrepancy occurs between the simulation results and the experimental data near the middle of the larger magnets. Note the dot lines in Figs 4(a) and 4(b), not only are the magnetic fields weaker near the middle area of the larger magnets [7], but the directions of the forces are flipped from repulsion to attraction. A similar discrepancy occurs when unlike poles are placed face to face.

In order to confirm this phenomenon, we repeat the experiment with magnets of different sizes, consequently we affirm that the attraction appears between like poles and repellence appears between unlike poles for a pair of interacting permanent magnets with large different sizes, it seems that a localized area with reversed pole exists in the middle region of the large permanent magnet. Some examples are shown in Fig. 5(a) in which the smaller magnets could be attracted and fixed in the middle of the larger magnet’s pole faces, when like poles are set face to face. For contrast, we also provide the pictures in which attraction exists around the edges of large permanent magnets when unlike poles are set face to face, and in Fig. 5(b), the smaller magnets are repelled in the middle of the larger magnet’s pole faces.

Fig. 5.

Six groups of examples shown in (a) present attraction of like poles and two groups shown in (b) present repulsion of unlike poles. For every group, the left photo displays the normal phenomenon and the photos to the right show the special phenomenon.

2.2. Characters in interacting processes beginning with new permanent magnet and measurement of surface magnetic induction intensity

Further, we prepare several newly purchased commercial N35 Nd₂Fe₁₄B permanent magnets, aiming at eliminate contingency. We use a 𝜙40 mm × 3 mm large cylindrical permanent magnet A as a measured subject, which is interacted with a 𝜙6 mm × 8 mm small cylindrical permanent magnet B. We repeatedly measure the surface magnetic induction intensity of both two magnets before and after every step of interaction. The experiment set-up is shown in Fig. 6, whose magnetometer (BST100) has 0.1 mT minimum resolution.

Fig. 6.

Magnetic induction intensity measurement.

For the convenience of description, the middle points O on the large magnet A’s N and S pole surfaces are defined as point O _N and O _S, respectively. Similarly, we mark two fixed symmetrical points P on the two pole surfaces and respectively define them as point P _N and P _S. The point P is 15 mm from the point O and close to the large magnet’s margin. The schematic drawing is shown as Fig. 7.

Fig. 7.

Schematic drawing of chosen point on the surface of permanent magnet we concerned in the experiment.

Meanwhile, we record the magnetic induction intensities of middle point on the small magnet B’s pole surface. As same as the larger magnet, we define the middle points of the smaller magnet’s two poles are Q _N and Q _S, respectively.

The concrete measurement procedures are as follows:

Step 1. Measure new magnets.

Before the interaction between magnet A and B, we measure surface magnetic induction intensities of the pair of newly unpacked permanent magnet. (We cannot promise no effect on the magnetism from the circumstance during transportation and storage, but it won’t affect the conclusions.) The distribution of magnetic induction intensities on the large magnet A’s surface appears weaker in its middle and stronger near around its margin. All values on N pole surface are positive, the minimum value is +43.4 mT, which belongs to the middle point O _N. Induction intensity at the point P _N is measured as +85.8 mT. Induction intensities of S pole surface are negative whose absolute values are the same as the values of N pole.

The values on the point Q _N and Q _S of the small magnet B are measured as +335 mT and −335 mT, respectively.

Step 2. Make S pole of the small magnet B (Q _S) face and move towards the middle point O _N of the larger magnet A’s N pole surface until they contact, and keep them touching for 30 seconds.

Always attraction appears in the interaction between two permanent magnets during the whole step 2. After we remove the small magnet B away, the remeasured magnetic induction intensities are not changed, both for two magnets.

Step 3. Make N pole of the small magnet B (Q _N) face and move towards the middle point O _N of the larger magnet A’s N pole surface until they contact, and keep them touching for 30 seconds.

Comparing with the situation in step 2, a new particular phenomenon of stronger attraction appears in the interaction between two permanent magnets during the whole step 3. This time, after we remove the small magnet B away, the values at the point O _N and O _S are respectively changed to −61.8 mT and +44.5 mT, and the values at the point P _N and P _S remain ± 85.8 mT, respectively. The values at Q _N and Q _S of the small magnet B still remain unchanged.

Step 4. Make the point Q _S face and move towards the middle point O _N and keep two magnets touching for 30 seconds.

After the operation in step 3, now the S pole of the smaller magnet is repelled against the middle point of the larger magnet’s N pole surface during the whole interaction in step 4. Then we take them far apart and measure the magnetic induction intensities again. The value at the point O _N has a little change to −54.6 mT, while the point O _S still remains +44.6 mT. Other values we concerned remain unchanged, including those of the magnet B.

Step 5. Make the point Q _S face and move towards the middle point O _S and keep the two magnets touching for 30 seconds.

Being similar with the situation in step 3, there is attraction between the two like poles of the two permanent magnets with large different sizes. We take magnets far apart and remeasure the magnetic induction intensities, values at the point O _N and O _S are −60.0 mT and +66.1 mT, respectively. Other values we concerned remain unchanged.

Step 6. Make the point Q _N touch O _S, Q _N touch O _N, Q _S touch O _N and Q _S touch O _S successively, each touch for 30 seconds, and repeat these operations several times (≥5).

In this step, when original like poles of the permanent magnets face to face, they attract to each other; when original unlike poles face to face, they repel against each other. We achieve the permanent magnet pair which has the special interaction mentioned in experiment 2.1. The magnetic induction intensities at the middle point O _N and O _S, respectively on the N and S pole surface of the large magnets, maintain −68.1 ± 3 mT and 68.0 ∓ 3 mT. The values at the peripheral point P _N and P _S remain unchanged as beginning, so do the points Q _N and Q _S of the magnet B.

For comparison, we take another pair of new magnets and make small magnet B’s Q _N and Q _S interact with points P _N and P _S of large magnet A as the same processing as the above steps. The result is that the induction intensities at O _N and O _S are almost unchanged, but the values at P _N and P _S are changed as the same trends as the above experiment step, except that their numerical values are different from the O _N and O _S’s. Figure 8 plots the magnetic induction intensities changed with every step. It is observed that local magnetization state of the large magnet can be reversed by the external magnetic field produced by the small magnet. We also discover in the experiment 2.2 that this special phenomenon can happen near both the middle and the peripheral areas of the large permanent magnet.

Fig. 8.

The changes of magnetic induction intensities with every step at (a) point O _N and O _S interacted with point Q _N and Q _S, (b) point P _N and P _S interacted with Q _N and Q _S. Other values of magnetic induction intensity remain unchanged, which are not plotted.

3. Analysis and explanation

For a block of permanent magnet working in an open magnetic circuit, due to the effect of demagnetizing field, the working point of a permanent magnet on its demagnetization curve is lower than B _r and defined by a load line, whose slope is $P_{C}=\frac{B_{D}}{{\mu}_{0}H_{D}}=\left(1-\frac{1}{N}\right)$ , also called permeance coefficient, derived from these equations when external field H _o = 0: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{D}={\mu}_{0}M\left(1-N\right)\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{D}=H_{0}-\mathit{NM}\end{eqnarray}$ (2) where N denotes the demagnetizing factor depending only on the magnet’s dimensions which is a constant; M is the magnetization intensity. There will be a unique working point for the magnet defined by its intercept with the demagnetization curve as shown in Fig. 9 [9].

Fig. 9.

Change in working point with load line slope.

Fig. 10.

Comparison of the magnetic flux distributions for two pairs: (a) Two magnets with large difference in permance coefficients or sizes; (b) Two magnets with the same permance coefficients or sizes. The top schematic drawings of each figure are solid models; the bottom ones show 3D FEA results with magnetic flux distribution [12].

Dynamic operation of a magnet in normal service results in movement of its working point, when the operation of a magnet at a point beyond the knee of a demagnetization curve, it represents an irreversible change. The magnet therefore follows the path of a minor B-H curve in Fig. 9. For example, the magnet works initially above the knee at point D. A change in permeance coefficient (load line slope) then shifts the magnet to working point E, below the knee of the curve. This change is irreversible, so even when it is then removed, the magnet returns to working point F instead of D along a minor B-H curve called recoil line. Subsequent application and removal of the same load line disturbance will cause the magnet to operate reversibly along this minor curve, recoil line, between E and F [9]. Consequently, the movement of working point depends on the demagnetizing factor N and the external field H ₀, high values of N and H ₀ can make working point move to the third quadrant.

Fig. 11.

Surface demagnetizing factor with respect to the diameter of large cylindrical magnet.

Always, N < 1 and the permeance coefficient $\left(1-\frac{1}{N}\right)$ is negative when the external magnetic field H _o = 0. We use an APP named Quadrant Magnetics Calculator to estimate the permeance coefficient of a permanent magnet in open magnetic circuit [10]. By comparing the magnets used in our experiment, the small magnet has a much higher load line or working point on the B-H curve, and the large magnet has a much lower load line or working point beyond the knee on the demagnetizing curve [11]. It can explain why large magnets’ surface magnetic induction intensities can be changed by the external magnetic field, which cannot happen to small magnets in our experiment.

Fig. 12.

Irreversible movements of working points at different positions of permanent magnets in experiment 2.2, the working points will stay on the recoil lines after the effect from the demagnetizing field. Black arrows for point O, blue arrows for point P and red arrows for point Q.

Figure 10 shows the magnetic flux distributions when a pair of permanent magnets interact. The FEA results are offered by a reviewer of this paper. For the magnet pair with large difference in permeance coefficients, even the magnets have like poles facing each other, the flux direction at the two magnets interface in Fig. 10(a) is dominated by the direction of the high permeance coefficient P _C magnet, so the repelling force is overcome by the attracting force and the magnetic polarity of localized area in the large magnet are irreversibly reversed. When permanent magnets have the same permeance coefficient due to the same size, the flux direction tends to go along the interface, as shown in Fig. 10(b).

In Fig. 8, we also observe that there are different amplitudes of induction intensities’ variation between the point O and P during the interactions in experiment 2.2. Most permanent magnets cannot be magnetized uniformly on account of demagnetizing field. Some researchers have demonstrated theoretically that only ellipsoid magnet could be magnetized uniformly [13,14]. Now if we don’t pay attention to a whole block of permanent magnet, instead, to the different local positions on the large permanent magnet such as the point O and P, there are different demagnetizing factor N _d. We calculate the N _d with respect to different positions inside a cylindrical permanent magnet using magnetic charge model [15], the detail calculations are shown in Appendix. We plot the demagnetizing factor N _d with respect to diameter of the 𝜙40 mm × 3 mm cylindrical permanent magnet shown in Fig. 11. Obviously, demagnetizing factor is higher in the middle area. Hence, N _d−O > N _d−P, so that the slope of point O’s load line is lower than the point P’s. That is the reason why the induction intensities’ variation at the point O is larger than the point P.

After the interactions between the small and large magnets in the experiment of Section 2.2, the working point is moved on the recoil line in the third and fourth quadrant (where B _m is reversed), as shown in Fig. 12. Therefore, it happens that localized area in the large permanent magnet with lower permeance coefficient can become reversed pole during the interaction between permanent magnets with large different sizes and, consequently, different permeance coefficient.

4. Conclusion and future work

This paper reports a special phenomenon we have discovered in our research work. When the size of one magnet is much larger than the other, the smaller magnet's north (south) pole can be attracted to the larger magnet’s north (south) pole surface. Subsequently, when opposite poles are faced to face, the smaller magnet could be repelled near the region attracted just before of larger magnet, because the magnetic polarity in this region is reversed and fixed. The reason for this phenomenon is the movement of load line or working point on demagnetization curve when a permanent magnet is effected by external field. If the slope of load line is low enough in the second quadrant when the external field is zero, especially much lower than that of knee point, the working point may move into the third quadrant, and stay on a recoil line in the third and fourth quadrant when this permanent magnet keeps working. Besides, we find that the equivalent magnetizing current model has a limitation to describe the interacting force between permanent magnets with different sizes.

By using this special newfound characteristic of permanent magnets, some mechanical systems could be designed with magnets having different sizes to realize such functions as limit control, fixation, and launch. According to the behaviors of the magnetic force curves shown in Fig. 4, the combination with some different sized magnets could produce a multi-stable potential function in three dimensions. In such a potential function, the depth of potential wells and the height of potential barriers can be modulated by changing the magnets’ relative positions and postures. Applying our finding in this classical field to more technological fields such as engineering control and energy harvesting can provide valuable and significant improvements in those respective topics.

Also, we will try to analyze the detailed effect of the parameters such as sizes of permanent magnets and interacting time on generating this phenomenon. Meanwhile, the distributions and postures of magnetic domains near the boundary of reversed area need further investigation.

Fig. 13.

Mathematical model of demagnetizing fields inside cylindrical magnet.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (grant No. 51675370).

Besides, we greatly appreciate to the reviewer of our manuscript, the reviewer’s comments, suggestions and the FEA figures make lots of contributions to this paper, which is helpful to improve the quality of our paper.

Appendix

We assume that the demagnetizing field is produced by the free micro magnetic charges distributed uniformly on the pole surface [15], and this demagnetizing field produced by the charges in the micro area dS of the pole face is (A.1) $\begin{eqnarray}\displaystyle dH_{d}={\displaystyle \frac{1}{4{\pi}{\mu}_{0}}}{\displaystyle \frac{dq_{m}}{t^{2}}}, & & \displaystyle\end{eqnarray}$ where dq _m = σdS, σ = 𝜇₀ M is the magnetic charge area density, and t denotes the distance from dS to point P (x, y, z) inside the magnet, as shown in Fig. 13, M is the remanent magnetization intensity. The fields produced by one circular pole face are calculated as ((A.2)) below. Hence, the demagnetizing fields in cylindrical are expressed as ((A.3)): (A.2) $\begin{eqnarray}\displaystyle H_{d}^{\prime }(r,z) & = & \displaystyle {\displaystyle \frac{{\sigma}}{4{\pi}{\mu}_{0}}}\int _{0}^{R}{\rho}d{\rho}\int _{0}^{2{\pi}}{\displaystyle \frac{zd{\theta}}{\left[({\rho}\cos {\theta})^{2}+(r-{\rho}\sin {\theta})^{2}+z^{2}\right]^{3/2}}}\nonumber\\ \displaystyle & = & \displaystyle MN_{s}(r,z),\end{eqnarray}$ (A.3) $\begin{eqnarray}\displaystyle H_{d}(r) & = & \displaystyle M(N_{s}(r,z)+N_{s}(r,L-z))=MN_{d}(r,z),\end{eqnarray}$ where L denotes the height of the cylindrical permanent magnet. We choose the pole surface of cylindrical permanent magnet (z = 0) to calculate the demagnetizing factors N _d with respect to different position on the surface, as shown in Fig. 11.

References

Cao

Tao

Cai

Jia

and Zhang

, Magnetic field analytical solution and electromagnetic force calculation of coreless stator axial-flux permanent-magnet machines, International Journal of Applied Electromagnetics and Mechanics 54 (2017), 93–105, doi:10.3233/JAE-160112.

Gysen

B.L.J.

Meessen

K.J.

Paulides

J.J.H.

and Lomonova

E.A.

, General formulation of the electromagnetic field distribution in machines and devices using fourier analysis, IEEE Transactions on Magnetics 46 (2010), 39–52, doi:10.1109/tmag.2009.2027598.

Kim

Choi

Koo

Shin

and Lee

, Characteristic analysis of tubular-type permanent-magnet linear magnetic coupling based on analytical magnetic field calculations, IEEE Transactions on Applied Superconductivity 26 (2016), 1–5, doi:10.1109/TASC.2016.2544948.

Iranna

Rao

and Soumendu

, Analysis of the magnetic field created by permanent magnet rings in permanent magnet bearings, International Journal of Applied Electromagnetics and Mechanics 46 (2014), 255–269, doi:10.3233/JAE-141776.

Zhang

Leng

Tan

Liu

and Fan

, Accurate analysis of magnetic force of bi-stable cantilever vibration energy harvesting system with the theory of magnetizing current, Acta Phyica Sinica 66 (2017), 220502, doi:10.7498/aps.66.220502.

Vučković

A.N.

Ilić

S.S.

and Aleksić

S.R.

, Interaction magnetic force calculation of ring permanent magnets using ampere’s microscopic surface currents and discretization technique, Electromagnetics 32 (2012), 117–134, doi:10.1080/02726343.2012.645430.

Medeiros

L.H.D.

Reyne

and Meunier

, About the distribution of forces in permanent magnets, IEEE Transactions on Magnetics 35 (1999), 1215–1218, doi:10.1109/20.767168.

Moss

S.D.

Mcleod

J.E.

Powlesland

I.G.

and Galea

S.C.

, A bi-axial magnetoelectric vibration energy harvester, Sensors and Actuators: A Physical 175 (2012), 165–168, doi:10.1016/j.sna.2011.12.023.

Campbell

, Permanent Magnet Materials and their Application, Vol. 63, Cambridge University Press, New York, 1996, pp. 88–96.

10.

http://magnetnrg.com/calculator/.

11.

Chen

Meng

and Fan

, Magnetic force equation for rare earth magnets and the effect of load line, in: The 25th International Workshop on Rare-Earth and Future Permanent Magnets and their Applications, Beijing, China, 2018.

12.

Chen

C.H.

, Private communication, July 2019.

13.

Zhao

and Chen

, Electromagnetism, 3rd edn, Higher Education Press, Beijing, 2011, pp. 382–401.

14.