Design of coaxial magnetic gear utilising a novel permanent magnet Halbach array structure

Ebrahim Afjei, Faculty of Electrical Engineering, Shahid Beheshti University, Rm 401, Velenjak, Tehran 1983969411, Iran. Email: e-afjei@sbu.ac.ir Abstract Mechanical gears are constrained due to wear and tear and call for lubrication. A magnetic gear, utilising permanent magnet (PM) arrays, facilitates the mechanical gear roles. It can withstand high‐speeds and large loads with zero friction. The magnetic gear is more efficacious on account of the least noise, performance, overload protection, and physical isolation between parts than conventional mechanical gears. The quantity of PMs in magnetic structures plays a principal role in its total price, which necessitates optimal arrangements for PMs. Halbach array provides a bigger torque density according to its one‐sided flux. In this research, proposing a novel Halbach array structure, leads to intensification in the one‐sided flux in a much sinusoidally waveform that consequently increases the magnetic torque. These all lead to gain higher torques in constant volumes of magnets. Besides, the magnetic torque ripple and the modulators’ losses are decremented. The new model is based on the variation of the magnetisation angles of the adjacent permanent magnet blocks, in the array. Various states of the magnetisation angles in the array are optimised based on a heuristic method. This is studied in the whole trigonometric range for magnetisation angles characteristics. The prototype is fabricated based on the optimised magnetic gear design and complies with the results.


| INTRODUCTION
Mechanical gears have a constrained life span due to wear and tear and they entail lubrication. A magnetic gear facilitates the same roles of a mechanical gear by employing permanent magnet (PM) arrays. It is superior over the mechanical on account of minimum noise, maintenance-free operation, overload protection (in that case, they slipper and when the load decrements to the nominal range, they return to the normal motion, but on the contrary the mechanical gear breaks.), and physical isolation between the components. Since there is no contact, technically the magnetic gear has a very long life and efficiency. At that, it can handle very high-speeds and large loads with zero friction. Mechanical gears are widely utilised to increment the speed of the initial actuator according to the loads demand. This acceleration can additionally be accompanied by an incrimination (high-speed ratio in wind turbine generators) and by decrementing it (vehicles and propulsion engines). With the contrivance of high-power PMs, it was possible to develop permanent magnetically based topologies that were operationally functional. PMs have advantages over electric magnets because they do not call for any energy input and cooling power; however, electric magnets are controllable. Despite their lossless characterisation, magnets have a good performance at room temperature, except for superconductors [1].
The basis of magnetic gear work is based on the magnetic fields generated by PMs and their passage through the modulators. The magnetic rotors are composed of different numbers of magnets, fixed on a yoke. Naturally, these rotors will produce different magnetic fields on the contrary orientation. To create effective torque and rotation of rotors, a field is needed in front of these rotors that are of the same order as the rotor's field. This is achieved by the soft magnetic components that fit between inner and outer rotors. Soft magnetic components, which are usually made of laminated steel or soft magnetic composite (SMC) materials, known as modulators, generate harmonics from the magnetic field by modulating the magnetic field passing through them on the other side, where can be coordinated with the opposite rotor and can generate torque [2]. These space-time harmonics operate at different speeds, and this difference in speed between main harmonics and other harmonics is the substructure of the work of magnetic gear. The coaxial magnetic gear is first presented by Atallah in 2004 while the first magnetic gear was founded by Armstrong in 1901 [3].
The magnetic field of PMs is almost sinusoidal in the airgap. The distribution of these magnetic fields in airgaps depends on the radius of the airgap, the radius and thickness of the PMs, the magnetic properties of the PMs, the number of pair poles, the initial angle, the rotational speed of the rotors, etc.
Problems that have caused procrastination in large-scale magnetic gears can be attributed to the complexity of its functional structure, its low torque density, and its high torque fluctuations. Ergo, in this research, new strategies and structures are proposed to increment the torque density. The quantity of PMs in magnetic structures plays a consequential role in its total price, which necessitates optimal structural design for PMs. Halbach array provides more torque, comparing other structures according to its one-sided magnetic field [4], which in our precedent work, the angular magnetisations in PM guideways for high-temperature superconductors levitation are applied, and the maximum levitation force was achieved. The major work was focused on the perpendicular magnetic force, as the railway was a smooth track; so the magnetic force fluctuations were not studied. But, in magnetic gears, because of modulators, it is required to improve the magnetic torque ripples. Earlier studies conducted on PM arrangements, utilising Halbach arrays [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], where PM magnetisations rotate in an array of magnets, creating the magnetic fluxes where vary in a more sinusoidal pattern toward the conventional PM arrays [20][21][22]. In the whole mentioned references, constant magnetisations, derived by division of 360°to the number of pole-pair pieces. In Reference [7], a 2-D model is proposed to calculate the magnetic field in a coaxial magnetic gear with Halbach PM array, considering dimensional parameters: width and height of the modulators ring and the thickness of the outer rotor yoke. Also for the same block sizes in each rotor, the outer and inner rotors block width is swiped so that the impact on the torque is observed. In Reference [8], the PM sizes are considered unequal; this leads to lower torque ripple harmonics. In Reference [9], the PM shapes are unequal, leading the airgaps to be non-uniform, and thus, the gear is optimised based on the torque ripple caused by the uniform airgaps. However, the maintenance cost would increase. Also based on the eccentric airgap shaped magnets, a higher precision is needed in fabricating the magnets, which also leads to higher costs. The previous works concentrated mainly on the optimisation of the dimensional characteristics of the PM arrays, albeit the magnetisation characteristics affect the torque, which has the major effect on the magnetic field, is not observed. So the optimisations that are done so far are not comprehensive.
In this study, a novel Halbach array is proposed based on the conceivable magnetisation angles of PMs and optimised to maximise the gear transferring torque. The proposed optimum designs lead to a reduction in the total magnetic gear cost on account of the least PM consumption in a specific size and provide maximum gear torque. After this, in part 2, utilising the proposed heuristic optimisation method, various case studies, according to adjacent PM pole-pieces magnetisation angles, are studied and modelled relying on derived key equations. The proposed method employed for the optimisation is a special way according to the considered parameters for the simulation and is highly appropriate for this work. This is for its fast and accurate procedure, particularly for studies like this, consuming a huge amount of memory and time. In part 3, the optimisation process is demonstrated. In Section 4, the derived results are discussed. Finally, in Section 5, the prototype is fabricated according to the optimised magnetic gear design. The ideal, conventional, proposed and experimental results are represented and achieved satisfying results and conclusions would be made.

| DESIGN CONCEPT
To ponder the conventional Halbach arrays with the proposed angularly magnetised arrays, several magnetic gears with various magnetisation angles are considered. This attempt is to approach the ideal Halbach array. PM arrays are simulated with variant feasible commencement and the rotation angles of the magnetisation vector, as described in Figure 1, while in the conventional Halbach arrays, the magnetisation rotation angle is commonly clockwise or counterclockwise of 90 or 45 based on PM pieces per pole pair [4]. Accordingly, the proposed model results in angular magnetisation vectors. In this study, the counterclockwise rotations of PM magnetisations are considered for the inner rotor and the vice versa for the outer rotor according to the one-sided flux, effective for each rotor.
To achieve symmetrically balanced PM array, the relationship between the commencement and rotation angles and the number of PM blocks in an array is obtained (1) for the inner rotor and as (2) for the outer rotor according to one-sided flux side of the array.
where according to Figure 2, α i and α o are the magnetisation angles of inner and outer rotors first PMs in the array, β i and β o are rotation angles of inner and outer rotors, respectively, at the trigonometric coordinate system, resulting in the adjacent PM magnetisations, and n i and n o are the number of PM blocks for inner and outer rotors, respectively. In this study, the modulator components are considered as the fixed part. So, there are two rotors: the inner and outer rotors. Because the transferring torque at different rotational speeds is correlated with the number of PM pole-pairs of the outer rotor and the number of paired harmonic poles of the desired space, the number of outer rotor pair poles should be equipollent to n s -p i as the relation, Equation (3), holds.
where p i is the number of pole pairs for the inner rotor, p o is the number of pole pairs for the outer rotor, and p s is the number of pole pairs for the fixed loop.
Regarding the largest harmonic component, the general relation of velocity in the magnetic gear can be shown as Equation (4): In the case of fixed intermediate modulator rotor: The considered coaxial magnetic gear has a gear ratio of 5:2. Seven pole pair steel pieces are inserted between a five pole pair outer rotor and two pole pair inner rotor. Each pole pair has consisted of an angularly magnetised Halbach array. For this work, arc PMs are employed to fit the structure and detract the air gap. Modern high power Neodymium-Iron-Boron (Nd 2 Fe 14 B) PMs with high coercivity, remanence, and relatively square hysteresis loops [23] are utilised for PM gears design. The dimensions and specifications of the samples studied in this section are shown in Table 1.
PMs are considered at full pace. At different boundaries, θ h , θ l , and θ m are, respectively, the angles of high-speed magnets, low-speeds, and modulators. The boundary conditions for the considered magnetic gear are shown in Figure 1. Initially, the following postulations should be considered: � The soft magnetic components (low-speed core, high-speed core, and modulator components) are assumed to be ideal (μ¼∞) and are considered as boundary conditions. � The end effects are ignored. � PMs have magnetised angularly, and their relative magnetic permeability coefficients are assumed to be (μ r ¼ 1.05). � The eddy current is ignored in the optimisation and later on, for the final model, its effect is considered.
Therefore, magnet areas (I, V), areas related to airgaps (II, IV), and the slot regions are as (III), which are represented by i (0<i < n s ). The magnetic vector potential relation is employed to model the problem. Laplace equations are utilised for airgaps and modulator slots, and the Poisson equation is employed for PM regions [24] for the convention.
where A → is the magnetic vector potential, M → is the magnetisation vector, and B r is the remanent flux density.

Magnetic field calculation
To design the proposed structure, applying the boundary conditions, in the finite element method for the optimisation, it is needed to calculate the magnetic field; thus, the magnetic torque will be yielded. The equations are in vector so the magnetic flux density is modelled in the areas. Consequently, they may be applied either for the conventional and the proposed Halbach array. The equation for boundary conditions and areas of high-speed rotor PMs is as follows: The boundary conditions are expressed as: F I G U R E 2 Array of permanent magnets with symmetry axis, α ¼ 45, β ¼ 90 and n ¼ 6 The complete solution of Equation (10) can be expressed as Equation (13).
such that: The A n I and C n I constants are obtained from the following relationship: 2 ; θÞ:sinðnθÞ:dθ In Figure 3 the magnetisation vector's tangential and radial components are plotted, respectively, for the example of outer rotor array. The arc magnets are initially considered as block magnets, to simplify the analytical calculations. But in the finite element calculations, the magnets are considered fully arced. The low-speed rotor magnetic field could be calculated, similarly.

Calculating torque
The electromagnetic torque can be calculated using the Maxwell stress tensor method. These relations are expressed in terms of the high-speed rotor as follows. The torque owing to the interaction of the radial component of the magnetic field with its tangential component causes the magnetic moment to be generated: where R m is the mean radius in high-speed airgap and L is the axial length of gear. The analytical relation of torque is as follows: The constants W n , X n , Y n , and Z n are given as follows. A similar torque can be extracted for the low-speed rotor.

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The derived equations are employed in the next section to optimise the proposed model, as the input model.

| OPTIMISATION PROCEDURE
In this study, it is objective to have a thorough vision about the impact of magnetisation angles in PM gears. Thus, the whole range for inner and outer rotor rotation angles should be investigated, and PM block numbers in the desired confine are considered. So, a huge amount of cases may be considered, and the simulations may be done. This calls for lots of calculations, memory, and time for that. For this intent, a heuristic optimisation method is proposed that acquires the optimum structure with the desired precision. Accordingly, it has a high accuracy in the results.
The magnetic flux density is calculated via 2D Finite Element Method (FEM) simulation, implemented by COM-SOL Multiphysics 5.4 LiveLink with MATLAB. Therefore, the different magnetisation structures of the PM pieces in the Halbach array are determined from the optimisation algorithm, and then based on this information, the corresponding 2D finite element simulations will be performed at high speed and accuracy. Then, the output performance results of the magnetic gear are referred to the optimisation algorithm, and the results will be analysed. This process will occur until the end of the algorithm iteration, and the information will be changed quasicontinuously.
The gear ratio, the PM gear block numbers for inner and outer rotors, n i and n o , respectively, which causes the different PM height and width, h & w, the rotation angles for the inner rotor, β i ; and outer rotor, β o , and the angle step, ε, are considered as the optimisation parameters. According to these parameters, PM gear structures would be simulated, and the outer and inner rotor torques, T o and T i are calculated. The method proceeds to the whole determined ranges extent for β i , β o , n i , and n o are covered, so that, the preliminary optimum points are accessed. Then, the angle step is minimised, and the process is reiterated for the neighbour of the optimum points to achieve the desired precision. In other words, the considered constraints are Equations (1) and (2) and the input parameters of β i , β o , n i , and n o variation ranges are in 0 to 360°and 2 to 11, respectively. The optimum cases were for block number ranging from 2 to 5. Through this heuristic iterative method, the optimum PM gear arrangements with the maximum torques are obtained. The optimisation method employed in this study is based on the sensitivity analysis of the structural parameters to inputs and direct results. Thus, the outputs are exported based on the inputs. According to the heuristic proposed method, the optimum points are opted out. The opted points are the peak points, similarly to the Pareto front graphs. As a change in dimensional parameters lead to a change in volume and weight, and in this paper, it was desired to increase the magnetic torque density in constant volume and weight; the effective non-dimensional parameters like the magnetic flux density of the magnets and their magnetisation angles are considered. The proposed optimisation method is represented in Figure 4.
As the optimum cases achieved, to validate and compare with the conventional structure, the electromagnetic and loss characteristics as well as efficiency and the gear ratio are extracted in the next section. This comparison verifies the optimisation algorithm. To better evaluate the results, the values of performance characteristics are normalised.
The comparison between the conventional and the proposed model parameters is demonstrated in Table 2.

| RESULTS AND DISCUSSIONS
Various simulation case studies have been performed. The summary of the optimisation results is shown in Figure 5. Where the normalised torque is plotted versus the outer rotor magnetisation rotation angle, β o , and inner rotor magnetisation rotation angle, β i .
Thus, the maximum structures could be extracted and studied. As shown in Figure 5, the torque variations resulted from the magnetic flux density energy conversion are collaborated with both, inner and outer rotor rotation angles. The whole range of inner and outer rotor rotation angles is F I G U R E 3 The magnetisation vector's tangential and radial components for the example of outer rotor array covered, and the possible states are studied by employing matrix permutation demonstrated in Figure 5.
From a general point of view, the torque variations with the inner rotor rotation angle is much smoother than the outer one. This demonstrates that the sensitivity factor for the outer rotor rotation angle is higher since the magnetic flux variations in the airgaps are sensed higher than the other areas, and the outer rotor has a wider surface, and it covers a major airgap in comparison with the inner rotor. Besides, the outer rotor's pole piece number is closer to the modulator piece number, compared with the inner rotor.
Also, it may be concluded that for most cases, the maximum structures appear in lower, as well as almost in 200-250°for the outer rotor magnetisation rotation angle range. This is owing to the sinusoidally one-sided flux pattern resulted from the mentioned magnetisation angles. As can be seen in the figures, the optimal structure is with a 24°angle for PM dimensions in external rotor magnets (15 magnets) and 90°for PM dimensions in internal rotor magnets (4 magnets), having a wider range of maximums or optimum points. The structure with the most number of PM block numbers has the least losses. As the number of PM block increments, the losses fall, but it has a saturated state because the magnetic flux density's RMS value scales down and implicates more different angularly magnetised blocks, which would result in higher intricacy and cost of PM gear's fabrication process. Also as it is pointed out in Table 1, the modulators and their slot angles are equally 25.7°according to Reference [6]; thus, if the PM segments decrease in size more than a level, the magnetic flux density, in some times, does not pass through modulators, and PMs see only the modulators slots. This leads to higher leakage flux in these cases. As a consequence, for future studies, the modulator slot angle may be varied also, through multi-objective optimisation. Additionally, as the number of blocks increases, the PM sizes are smaller so they have lower magnetic flux besides the leakage flux between the adjacent magnets increases also, but it also depends on the magnetisation vector for each PM. Accordingly, the magnetic flux density's RMS falls. These all lead the torque to a trade-off. The curves with smoother modifications are better suited for construction because even a slight error in the construction of magnets with magnetisation angles extracted from graphs with high fractures and discontinuities may result in torques with significant disagreement with the derived results. On the other hand, it can be seen that in some parts of the peaks, the torque is almost saturated, which can be used to approximate structures that have almost identical torque with low disparity, but the cost and complexity of constructing magnets with those magnetisation angles is less. This is not, of course, the case for magnets being manufactured by the magnet manufacturer or large enough external magnetic field (considering the magnetisation curve for each material) for demagnetising magnets. For little torque structures, this is owing to the concentration of the one-sided flux on also the non-desired side of the array, for inner and outer rotors.
As shown in Figure 5, structure with 15 magnets for external rotor with n o ¼ 3 and four magnets for internal rotor with n i ¼ 2 and α o ¼ F02D203.4°, β o ¼ 293.4°, α i ¼ 155.1°a nd β i ¼ 229.9°is the optimal structure having external torque of 1.29 Nm and internal torque equal to 0.53 Nm. Similarly, for the other values of n, the angles β o , β i , α o , α i , and the schematic PM gear array structures, which result in the maximum torques, are presented in Table 2. For the magnetic gearboxes with different ratios, the arrays are repeated and proportional to the internal and external rotor ratios, forming pole pairs.
According to the optimisation results, the optimum PM structures are those wherein PM magnetisations have a more sinusoidal magnetic flux pattern, that is a consummate arc in a moiety of PM array cross-section. This may be further considered comparing Figure 6a -305 blocks. The results show that the outer rotor torque for the array of Figure 6a is almost 50% higher than the array of Figure 6b.
The optimised simulated magnetic gear with angular Halbach PM array is shown in Figure 7. The magnetic flux density norm is shown in surface, and the colorbar illustrates its intensity range. Accordingly, the magnetic flux density is uniformly distributed.
The contour plot shows the magnetic vector potential, and the arrow surface demonstrates the magnetic flux density orientation. As can be seen, the mesh plot is fine enough to achieve the desired precision in results. The instantaneous inner and outer rotor torques are compared in Figure 8 for the conventional and optimised models. The torque ripple of the optimal model is 61.39% lower than in the conventional case. This is further discussed in Section 5.

| FABRICATED PROTOTYPE
The selected optimum magnetic gear design based on the angular Halbach structure proposed is fabricated and shown in Figures 9-11. The prototype is constructed of three components. The fixed on outer plane modulators, the high-speed inner rotor, and the secondary outer rotor. The non-magnetic and non-electric parts are constructed from the fiberglass material having higher mechanical and heat stability to achieve the minimum losses. The modulator components are laminated by the EDM machine with high accuracy to reduce the eddy current losses. The modulator components are fixed in place by the non-magnetic holders. The bearings are also located on the edges of the holders. The inner rotor, located in the area middle the holder, is fabricated by employing optimum angularly magnetised PMs. The outer rotor has 15 angularly optimum magnetised PMs with a shaft out on the other side of the plane.
The ascertainment test shows that the maximum outer rotor torque is of 1.3 Nm. This complies with the optimisation results.
As can be seen in Figure 12, in lower primary speeds the losses for both, proposed and conventional models are less and about, but as the speed increases, the conventional model's efficiency decreases rapidly, on the contrary.
Albeit both structures experience the alike magnetic flux frequencies, according to higher magnetic flux passing through the modulators, higher losses eventuate. Eke, larger PM piece sizes, cause bigger losses.
The results of the fabricated model also confirm the optimisation results having a 0.7% error. The PM losses are almost 25% of the modulator losses. This is because modulators experience alternative flux with extreme frequency variations, having various types of harmonics. This is according to the different pole pair numbers in rotors. The larger the rotor core is, the larger it will not be saturated. Thus, for the comparison, the conventional model's rotor yoke that its size can be easily computed is considered 50% larger of the worst case of the proposed studied models that have the most magnetic flux density, for not to go saturated early. So, the optimal model is far better than the conventional model. Figure 13 compares the torque conversion ratios for ideal, conventional, proposed optimal and fabricated models. The losses cause the torque ratio to be lower than the ideal one. The torque conversion ratio for the conventional model decreases rapidly with increasing primary speed, which F I G U R E 8 The instantaneous inner and outer rotor torques for the conventional and optimised model F I G U R E 9 The fabricated prototype 306demonstrates the superiority of the proposed optimal model in this regard, as discussed for the efficiency. The results from the constructed model, with a discrepancy of 0.9%, match the computational results.
The lower losses, leakage fluxes, and improvements in magnetic flux density distributions lead the resulting torque density to be higher (having desired harmonics amplitudes to approach the ideal gear ratio, in other words, it helps the performance of the modulators.). There are two types of losses in magnetic gear: iron losses and eddy current losses in PMs. As the magnetic gear is surface mounted, the core losses are only in modulator components, because the magnetic flux density in inner and outer rotor soft iron areas has no alternative component. The main reason for the better efficiency of the proposed optimal model than the conventional one is minimum core losses, while the optimised Halbach model with angular magnetisation vectors is considered.
Due to the rectification of the field on one side of the array and a smoother sinusoidal pattern, the density of magnetic fluxes decreases in modulators.
The magnetic flux density range in the angular Halbach state for modulators has different values as it passes the various places of the modulators, and its RMS value has a sharp decline. So in the conventional model, the modulators are saturated with the magnetic flux density (almost 1.8 T), having maximum losses. The reason is that the magnetic flux enters these areas angularly. This decrease in amplitude of the magnetic flux density in these areas decreases the core losses. So, it is almost 30 W in conventional and 4.5 W in the proposed model (for the maximum simulated speed).
According to Figures 14 and 15, which shows the losses distribution in the modulators and PMs of the conventional PM arrangement and the new Halbach array, respectively, the amount of losses in the parts of the new design has been significantly reduced. Also, the losses in the components of the Halbach model has occurred more locally. Another reason is that when segmenting, for example, in the external rotor, instead of 10 large magnets (large cross-section and large perimeter length), 15 magnets were utilised, and the crosssection was reduced, as in Reference [5].
Accordingly, when the alternative flux (created by PMs and alternative modulators reluctances) passes through the PMs, the losses decrease extremely. As a result, losses due to eddy current in the magnets decrease as their numbers increase.
While maintaining the effective cross-section of the magnets, the number of pieces increases, and their dimensions become smaller. Of course, for the conventional model, for these reasons, a larger core is needed, which does not saturate, so the weight and cost increase again. Further, because the modulator flux is low, the magnetic flux changes in the magnets are also sensed. Additionally, the magnet magnetisation vector is angularly charged; thus, when a segment departs from or approaches a magnet, those locations cause the most losses because the magnetic flux has to be deflected and angled.
But by applying the angular Halbach model, the magnets themselves create these angular paths for magnetic flux. The number of deviations from the magnetisation vector of the magnets decreases and reduces the eddy current losses. So as the magnetic flux in the core decreases, the amount of core and the cost are reduced.