# A New Analytical Torque Formula for Axial Field Permanent Magnets Coupling

Abstract: In this paper, we present a simple and accurate analytical expression to compute the torque of axial-field magnetic couplings. The torque expression is obtained by solving the three-dimensional (3-D) Maxwell equations by the method of separation of variables. Here, we adopt the assumption of linearization at the mean radius, the problem is then solved in 3-D Cartesian coordinate (we neglect the curvature effects). To show the accuracy of the torque formula, the results are compared with those obtained from 3-D finite-element simulations and from experimental tests. As the proposed formula needs very low computational time and depends directly on the geometrical parameters, it is used for a design optimization using multiobjective genetic algorithms.

## Summary (2 min read)

### Introduction

- Analytical expression to compute the torque of axial-field magnetic couplings.
- It consists of two similar rotors facing each other.
- The electromagnetic computation of magnetic couplings is carried out using several methods.
- The main drawback of FE methods is the long computation time and the lack of flexibility.
- A new and purely analytical expression for the torque evaluation is then derived from the 3D solution.

### A. Magnetic force from the electrostatic-magnetostatic analogy

- Unlike s, the magnetic charge m doesn’t have any physical meaning.
- It is introduced for modelling purposes in which it usefully replaces some magnetic field sources (magnets, current carrying solenoids,…).
- Furthermore, expression (2) which uses Lorentz force in free space gives, for their coupler, the right values of the force along the x and y directions only (no other material than air in these two direction).
- Since ferromagnetic materials are present in the z-direction, (2) will not give the right value of the force and the authors have to use Maxwell stress tensor or virtual work methods.

### B. Magnetic field due the magnets of one rotor

- Fig.3 shows the problem to solve after linearization.
- The main dimensions of the linearized coupler are Due to the alternate polarity along the x-direction, only one pole is considered with anti-periodic boundary conditions along x.
- These conditions state that the tangential magnetic field components Hx and Hy are zero (iron boundaries).

### C. Equivalent surface charge density of the second PM rotor

- This dot product has to be performed on all the external surfaces of the magnet volume.
- Fig.6 shows a rectangular permanent magnet with a uniform magnetization in the z-direction.
- From (13), the magnet is then represented by two surface charge densities + and -.

### D. Torque expression

- The force is computed using (2) where the integration is performed on the surfaces carrying 𝜎+ and 𝜎−.
- According to the boundary condition (3), the tangential components 𝐵𝑥 and 𝐵𝑦 of the flux density are null on the charged surface 𝜎+ (at 𝑧 = ℎ𝑡), so the forces that contribute to torque (Fx and Fy) also vanishes.
- 𝑋𝑖 −𝑙𝑚𝑥+𝑋𝑖 𝑑𝑥 𝑑𝑦 (14) The variable 𝑋𝑖 in (14) corresponds, in cartesian coordinates, to the angular lag (load angle) 𝜑 between the two rotors of the coupling.
- Another useful quantity to compute is the flux over a pole surface.
- The thickness of iron yokes can be determined using flux conservation law.

### III. EVALUATION OF THE TORQUE FORMULA

- The authors analyze the accuracy of the developed torque formula whose results are compared to those issued from 3D FE computations (Comsol multiphysics®) carried out on the actual cylindrical coupling.
- The yoke thickness is choosen to avoid magnetic saturation.
- Hence, a linear model with a relative permeability value equals to 1000 is used for the ferromagnetic material so the computation time is reduced without any loose of accuracy.
- The corresponding experimental and analytical results also serve to evaluate the torque formula (17).
- If the authors add a supplementary harmonic (𝑁 = 3), the computation provides very accurate result with an error estimate lower than 2%.

### B. Comparison to experimental and 3D FE results

- The authors compare experimental measurements for the static torque with numerical and analytical computations.
- An analytical formula which is derived in [11] using a 2D analytical model (mean radius model and first harmonic approximation) is given by (21).
- This formula doesn’t take into account the edge effects.
- 9 show the results of the torque calculation for two airgap values (e=4mm and e=9.5mm) obtained by all the methods in use.
- This clearly shows the necessity to consider the radial fringing effects in axial field couplings.

### C. Curvature effects

- In order to address the limits of the analytical formula regarding the curvature effects, the analytical computation (linearized coupling) are compared to 3D finite element simulations (actual cylindrical topology) for several dimensions of the magnet.
- For a given air gap, the error introduced by the linearization assumption depends on the radial excursion Rout-Rin and on the mean pole pitch which is equal to (𝑅𝑜𝑢𝑡 + 𝑅𝑖𝑛)/2𝑝. (see Fig.2).
- But finally, the authors found that this single parameter λ led to the same conclusions.
- The first one uses 3D FEM to obtain the torque of the cylindrical coupling, also known as Two computations are performed.
- This corresponds to 504 combinations (252 for each topology).

### D. Optimization of the coupler using genetic algorithms

- Genetic Algorithms (GA) are widely used as a robust and effective tool in optimization problems.
- The objective T2 (Nm) is the torque for an air gap e=10 mm and the objective MPM corresponds to the total mass of the PMs.
- The optimization procedure uses 100 individuals evolving during 100 generations, but 50 generations are enough to reach a stable solution.
- Notice that many of the solutions of Fig.11 have checked by the 3D FE model and the error in the worst case is less than 3%.

### IV. CONCLUSION

- A new analytical expression to compute the torque of a PM axial field magnetic coupling has been derived.
- This expression has been obtained thanks to 3D magnetostatic analytical.
- By introducing a surface charge density, the torque computation used the electrostatic-magnetostatic analogy to evaluate the Lorentz force.
- The authors have shown that the proposed torque formula is very accurate and computationally very efficient.
- Thus, it has been used to optimize the studied coupler by a multiobjective genetic algorithm.

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##### Citations

54 citations

### Cites background from "A New Analytical Torque Formula for..."

...Overall, most of the 3-D magnetic field problems that have been recently solved by an analytical way are most often dedicated to the magnetostatic case [27]–[30], and very little attention is given to the 3-D eddy-current problems with moving conductors....

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45 citations

19 citations

### Cites background from "A New Analytical Torque Formula for..."

...As shown in [15] and [16], the curvature effects can be analyzed in an effective manner by considering a dimensionless number λ defines as the ratio of the radial excursion of the magnets R = R2 − R1 around the mean radius Rmean = (R1 + R2)/2 to the pole pitch τ λ = R τ with τ = π p Rmean....

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6 citations

### Cites background or methods from "A New Analytical Torque Formula for..."

...This paper presents a different type of analytic approach to that presented previously [7, 8, 12-17]....

[...]

...Although the approach presented here is for a Halbach AMC the presented approach could be extended to model AMCs that contain magnetic steel [7, 8, 12, 13]....

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...created a 3-D analytic model of a magnetic coupling by ignoring the curvature effects [12, 13]....

[...]

6 citations

### Cites background or methods from "A New Analytical Torque Formula for..."

...However, most of the time, MCs are classified regarding their magnetic flux geometries, which are usually classified as axial or radial [78]....

[...]

...proposed what is called as “ideal” AMC and RMC, with rotors made from arc-shaped magnets in Halbach arrays, and their torque density analyses are based on a new analytical formulation, including the curvature effects that were neglected in [78]....

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...In [78], it is proposed a new analytical formulation using a subdomain method and the torque density optimised by Genetic Algorithms (GA)....

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##### References

143 citations

### Additional excerpts

...Usually, the problem is solved under a two-dimensional (2-D) approximation which, in some situation like in axial field couplers, results in a 30% overestimation of the torque compared to three-dimensional (3-D) FE prediction [10]–[12]....

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121 citations

### "A New Analytical Torque Formula for..." refers methods in this paper

...For simplicity, let us consider, in free space, an electrostatic uniform surface charge density σs (C/m2), subjected to an electric field E [16]....

[...]

...From the magnetostatic point of view, it is usual to use an equivalent magnetic surface charge σm in A/m [7], [11], [14]– [16]....

[...]

84 citations

### "A New Analytical Torque Formula for..." refers background or methods in this paper

...Axial flux magnetic coupling prototype [11], [12]....

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...An analytical formula which is derived in [11] using a 2-D analytical model (mean radius model and first harmonic approximation) is given by (21)....

[...]

...7) (more details can be found in [11] and [12])....

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...In [11] and [12], the authors have constructed a prototype axial field coupling and developed a 2-D formula to evaluate the torque....

[...]

...From the magnetostatic point of view, it is usual to use an equivalent magnetic surface charge σm in A/m [7], [11], [14]– [16]....

[...]

77 citations

### "A New Analytical Torque Formula for..." refers background in this paper

...3-D analytical models for magnetic couplings have been proposed in the literature [6], [13] and [14]....

[...]

66 citations

### "A New Analytical Torque Formula for..." refers background in this paper

...3-D analytical models for magnetic couplings have been proposed in the literature [6], [13] and [14]....

[...]

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