Torque sensitivity analysis for triple ‐ speed coaxial magnetic gear using finite element method

The authors introduce a new type of triple ‐ speed coaxial magnetic gear (TSCMG) and analyses its operation with sensitivity analysis. The effect of various factors is evaluated on TSCMG performance, including the yoke thickness, permanent magnet (PM) thickness, number of pole pairs (gear ratios), axial length, pole ‐ arc coefficient, PMs and air gap length. At first, a preliminary design of TSCMG is made, and then its optimal parameters are selected using sensitivity analysis based on the finite element method (FEM). Then, TSCMG is simulated by applying both preliminary and optimal design parameters. The flux density distribution and rotor torque within TSCMG are extracted. Analysis results show that TSCMG can offer higher pull ‐ out torque and lower maximum flux density at the edges and corners of TSCMG. Finally, the performance of the designed TSCMG is validated with a prototype.


| INTRODUCTION
One of the important components of industrial applications is gearboxes. They are applied in several applications, such as electric vehicles, crashers, the wind power industry, and electropumps. In the above mentioned applications, mechanical gears play an important role. The mechanical gears have some disadvantages, such as acoustic noises and vibration maintenance problems, and the need for periodical lubrication [1][2][3]. A new type of gear is introduced, namely magnetic gears (MGs), which overcomes the shortages of mechanical gears [4]. A magnetic gear is a frictionless device for transferring speed and torque, and provides more advantages over mechanical gears [5].
A conventional magnetic gear includes two PM arrays mounted by the inner and outer rotors. So, it can transform torque and speed via interacting magnetic fields of the rotors. This characteristic of the magnetic gears brings out some special advantages that includes high reliability, low noise, protection against overload, isolation of input and output shafts, and reducing the maintenance requirement [2,6,7]. Various kinds of magnetic gears have been introduced in the literature, such as linear magnetic gears [8], trans-rotary axial magnetic gears [9,10], and coaxial magnetic gears [11,12]. Each and every type has advantages and disadvantages.
Recently, numerous papers investigated the effect of design parameters to improve the torque in double-speed magnetic gears. The effects of the yoke and magnet thickness have been studied on the maximum static torque in [13]. The Halbach arrays were also used to optimise the wideness of the modulation loop core, the height of the modulation loop and the outer yoke thickness of CMG [14]. The optimisation of relationships between the design parameters has been investigated for the maximum torque in [15]. Further, in the other study, the effects of design parameters have been examined on optimising the torques [16]. The effects of design parameters including the number of pole pairs in the inner and outer permanent magnets and ferromagnetic equipment, the ferromagnetic geometry and the PM thickness of gear have been investigated for optimal torque and gear performance in [17]. The effect of ferromagnetic poles is also studied in [18].
The electronic-continuously variable transmission (E-CVT) system has been presented and optimised in [19]. The given system, i.e. E-CVT, consists of two rotors and two stators, in which one of the rotors and stators are wound. Moreover, four coaxial gears, CMGs and magnetic conductive rings (MCR) from torque and demagnetisation prospects have been investigated using the FEM in [20]. A new PM machine with double-layer PM excitation, named as a double-layer magnetic-geared permanent magnet (DL-MGPM), has been presented in [21].
The present study has focused on analysing the design parameters of TSCMG to achieve optimum torque and flux density. It is worth mentioning that the design consideration of TSCMG is completely different from its double-speed gear counterpart. TSCMGs are useful devices in many industries, especially in wind turbines, and cause it to generate more power. The third rotor in TSCMG improves the efficiency of wind generation. So, to the best of our knowledge, the main contribution of this paper is presenting a new configuration for magnetic gears and sensitivity analysis to the optimal design of TSCMG parameters. The innovative part of this paper is constructing a prototype of TSCMG and proving how the middle rotor can be connected to the load to generate artificial wind.
The paper is organised as follows: In Section 2, the principle of TSCMG is described. Then, sensitivity analysis of TSCMG is performed using FEM in Section 3. The optimisation and final structure selection are discussed in Section 4. Validation of designed TSCMG is done by a fabricated prototype in Section 5. Finally, the conclusions of the present work are discussed in Section 6. Figure 1 shows an outline of the proposed TSCMG. The radial and tangential components (B r and B θ ) of magnetic flux density produced by each PM rotor can be written as follows [22]:

| PRINCIPLE OF TSCMG OPERATION
where p is the number of pole pairs in the permanent magnet rotor, n s is the number of ferromagnetic pole-pieces, ω r and ω s are the rotational velocities of the permanent magnet rotor and the ferromagnetic pole-pieces, respectively. b rm and b θm are Fourier coefficients for the radial and tangential components of the flux density distribution, without the ferromagnetic pole-pieces, respectively. λ rj and λ θj are Fourier coefficients for the modulating functions related to the radial and tangential components of the flux density distribution resulting from the introduction of the ferromagnetic pole-pieces, respectively.
From Equations (1) and (2), the number of pole pairs in space harmonics distribution of flux density is given by: From Equation (4), the rotational velocity of the flux density space harmonics can be determined as follows: where Ω h is the speed of the inner rotor. By substituting the amplitude of high-order harmonics in Equation (1) and in the case where m ¼ 1 and k ¼ À 1, the rotation speed in the middle rotor is obtained as follows: where p h is the number of pole pairs of the inner rotor. The speed of the outer rotor is given by: Since the suggested structure consists of three rotors, there are two transformation ratios as follows: where p m is the number of pole pairs in the middle rotor, Ω m is the speed of the middle rotor, and Ω l is the speed of the outer rotor. Also, the number of pole pieces in the given gear is obtained based on Equations (9) and (10): where p l is the number of pole pairs in the outer rotor. As noted earlier, this rotor has three different torques, which are as follows: where j ¼ {h, m, l} ( inner, middle, and outer rotors ). L ef is the effective axial length called stack length. R is the air gap radius. Based on the above-discussed equations, the preliminary design of TSCMG was made. It is noted that the proposed TSCMG includes three (inner, middle, and outer) rotors with different torques.

| SENSITIVITY ANALYSIS USING FEM
The structure of the proposed gear is a new type, and the design parameters of TSCMG have not been analysed previously. Therefore, it is necessary to investigate the effects of design parameters on the torque, flux density and other variables. In this section, first, a preliminary design of TSCMG is made and its parameters were listed in Table 1.
Then, the effective parameters on the TSCMG operation are evaluated using sensitivity analysis, and the optimal value of parameters is selected. The study parameters are the yoke thickness, PM magnets, pole pieces, number of pole pairs (gear ratio), axial length, pole-arc coefficient of PMs and the air gap length. Figure 2a shows the effect of the yoke thickness on the flux density distribution. As shown in Figure 2a, the flux density at the edges is in the range of 4.5-5.3 T, which is not desired for TSCMG. The minimum and maximum torque in the inner, middle and outer rotors of TSCMG are illustrated in Figure 2b, which shows that the yoke thickness has no effect on the maximum torque. -407

| PM thickness
The magnet thickness has the greatest effect on the torque and flux density. Therefore, they should be analysed separately for inner, middle and outer rotors. In order to evaluate the effect of PM thickness on the TSCMG operation, the PM thickness is varied from 4 to 12 mm. In other words, when increasing the PM thickness, the thickness of pole pieces gets reduced due to the constant outer rotor's diameter. As shown in Figure 3, the torque and maximum flux density are affected by the variation in the PM thickness. The flux density distribution within the edges of TSCMG is depicted in Figure 3a, which is varied between 4.2 and 5.2 T. It is obvious that when increasing the PM thickness up to τ p ¼ 10 mm, the flux density increases, and the flux density decreases from 10 mm onwards. So, it can be said that the maximum flux density is achieved at a PM thickness of 10 mm. The outer, middle and inner rotor output torques are depicted in Figure 3b. The rotor output torque increases when the PM thickness is increased up to 10 mm, and after this value, the torque reduces. The maximum accessible output torque is 2200, 2100 and 1100 N.m for outer, middle and inner rotors, respectively. Some analyses have been performed to evaluate the effect of each rotor's PM thickness on the TSCMG operation, as shown in Figure 4. As shown in Figure 4a, the inner rotor torque gets increased when the magnet thickness is increased. However, the torque in the middle and outer rotors has no significant changes, and the flux density at the edges only varies marginally.
In this analysis, when changing the PM thickness of inner rotor from 6 to 10 mm, the torque of inner, middle and outer rotors increases to 1039, 1590 and 1717 N.m, respectively. The other study factor is variations in the PM thickness of outer rotor. To evaluate its effect, the PM thickness of outer rotor is changed from 6 to 10 mm, and its effect on the rotor output torque is shown in Figure 4b. As shown in this figure, the inner rotor torque is approximately constant, and the middle rotor affects the torque of the inner and outer rotors positively. In this analysis, the maximum achieved torque for inner, middle and outer rotors are 1222, 2214 and 2268 N.m, respectively.
The variation of PM thickness of middle rotor and its effect on the output torque is shown in Figure 4c. As can be seen in the figure, when increasing the middle rotor thickness, the PM thickness affects all the three rotor output torques. In this case, the maximum torque achieved for inner, middle and outer rotors are 870, 2021 and 2287 N.m, respectively.
Further, to investigate the effect of two rotors' PM thickness simultaneously, three cases were considered and studied. In case 1, the PM thickness of inner and middle rotors is varied, and its effect on the output rotor torques is shown in Figure 4d. In this condition, all rotor torques are increased, and the maximum torque achieved for inner, middle and outer rotors are 1443, 2258 and 2232 N.m, respectively. Figure 4e shows the rotor output torques when the thickness of both the middle and outer rotor are concurrently varied from 6 to 10 mm. In this case, the maximum torque achieved for inner, middle and outer rotors are 1077, 3919 and 3060 N.m, respectively, and when compared with the previous scenario, the middle and outer rotor torques have increased considerably.
The last case is changing the PM thickness of inner and outer rotors. According to Figure 4f, the torque of the middle and outer rotors increases by increasing the magnet thickness. These changes also affect the inner rotor, and its torque increases. The flux density at the edges changes negligibly.

| Number of PM pole pairs
The effect of PM pole pairs on the output torque is shown in Figure 5a. In this case, at first, the pole pairs in the inner, middle and outer rotors are specified as 1, 2 and 3, respectively. In other words, the gear ratios G r1 and G r2 of TSCMG are equal to 2 and 1.5, respectively, based on Equations (7) and (8). When the pole pairs are increased, both gear ratios are decreased. From Figure 5b, it can be concluded that the output torques is decreased by increasing gear ratios up to G r1 ¼ 9/8 and G r2 ¼ 10/9, and then they will be fixed.

| Axial length of the magnetic gear
The parameter that has effect on the operation of TSCMG is the axial length of magnetic gear called stack length. For different stack lengths of magnetic gear, the FEM analysis was performed, and its results are shown in Figure 6. It can be seen that the stack length affects the rotor output torques significantly. The increase of stack length leads to an increase in the output torques. At the optimum value

| Pole-arc coefficient
The pole-arc coefficient can be calculated as follows: The variation of K pi and its effect on the TSCMG torque is shown in Figure 7. As can be seen in Figure 7(a), the torque increases when the parameter α pi (shown in Figure 1) is changed from 20% to 65% in inner, middle and outer rotors, and then decreases. In this analysis, the minimum torque for inner, middle and outer rotors is 437, 904 and 961 N.m, and the maximum torque for these rotors is 1089, 1655 and 1841 N.m, respectively. Figure 7(b) shows the effect of K pi on the flux density at the edges and corners, which is in the range of 4.28-4.7 T. This figure verifies that the flux density increases proportionally with K pi .

| Pole-arc coefficient of PMs
The PMs are a cost-effective part of the magnetic gear. In order to reduce the PM cost, the resulting pole structure is applied to the proposed topology, as shown in Figure 1. Same the pole-arc coefficient of PMs can be defined as: To evaluate the effect of K Mi on the rotor output torque, its value is varied from 0% to 30% for the rotor. The maximum rotor output torque is achieved and shown in Figure 8a.
Based on this figure, it is observed that the output torque reduces proportionally to the increasing K Mi . For example, the middle rotor torque reduces from 1600 to 1003 N.m for 0% < K Mi < 30%. Flux density distribution within edges with K Mi changes is shown in Figure 8b. As shown in this figure, the flux density considerably reduces by increasing the pole-arc coefficient.

| Air gap length
The air gap has more effect on the output torque of gears. Variation of torque by air gap length has been illustrated in Figure 9.
As can be seen in the figure, when the air gap length is varied from 0.5 to 4 mm, the inner rotor torque decreases from 1148 to 581 N.m and for the middle rotor, it reduces from 1128 to 1094 N.m, while for the outer rotor, it decreases from 2446 to 2222 N.m.

| OPTIMISATION AND FINAL STRUCTURE SELECTION
As stated before, the aim of authors is to optimise the design parameters of TSCMG. In order to optimise the torque and flux density distributed on the edges and corners, the thickness of pole pieces in both inner and outer rings needs to be different. Moreover, it is found that the magnet thickness increases when decreasing the inner modulator thickness because the flux density at the edges will not increase using this technique. So, the magnet thickness has a tremendous effect on the torque. The middle rotor significantly affects the torque of inner and outer rotors and the flux density distribution on the edges. According to the above statements, the overall  Table 2.
To examine the effect of optimal parameter value on the TSCMG operation, two topologies of TSCMG are analysed based on FEM. At the first structure, TSCMG is simulated with preliminary design parameters, as given in Table 1. The flux density distribution within TSCMG is shown in Figure 10a, which indicates that the maximum flux density is 4.57 T, and it happens at the edge and corner of magnetic gear. The parameters of the second topology, called optimised topology, are extracted from the sensitivity analysis in Section 3. In other words, this topology is optimised concerning the previous one. Figure 10b shows that the flux density distribution along TSCMG, which indicates that the maximum flux density reduces to 2.06 T at the edge.
The rotor output torques shown in Figure 11 are for preliminary and optimised designs. For preliminary design, the maximum torques for inner, middle and outer rotors are 709, 1452 and 1917 N.m, respectively. In the optimised design, the maximum torques for inner, middle and outer rotors are 906, 1777 and 2408 N.m, respectively. It confirms that the torque in the optimised design has significantly improved by about 20% compared to the preliminary one.

| EXPERIMENTAL VERIFICATION
The optimised TSCMG with parameters listed in Table 3 is prototyped, as shown in Figure 12. There are subtle differences between the characteristics of constructed TSCMG and the simulated sample due to fabrication limitations. Figure 13 shows a layout of the constructed TSCMG in the laboratory, which includes three separate shafts on the inner, middle and outer rotors. The middle rotor shaft is rotated via a belt that connects the shaft to the load, as shown in Figure 13. The test bench consists of one motor, three torque meters, One of the important parts of the fabrication process is the middle rotor and its restriction. As shown in Figure 14, the middle rotor should connect to the belt to provide third and additional speed.
The flux density distribution along TSCMG is shown in Figure 15a, which reveals that the maximum value is 2 T and it is approximately equal to the simulated one. The torque of rotors versus time is shown in Figure 15b. As it is seen, the maximum torque in the inner, middle and outer rotors is 64, 86, and 116 N.m, respectively.
The torque-speed values for the curve of each rotor are measured here. The maximum torque of rotors is recorded and shown at different speeds in Figure 16. Here, the inner rotor speed is varied from 100 to 400 at 50 rpm in steps, and torques of rotors are measured. The torque speed shows that the torque decreases by increasing rotor speed. The comparison between simulation and experimental results is shown in  Figure 17. As expected, the experimental results verify the simulation ones, and just a subtle difference is observed between them, which are in the range of 4%-7%.

| CONCLUSIONS
A new TSCMG was introduced by the authors, and its sensitivity to parameter changes was investigated. Analyses showed that the yoke thickness has the least effect on TSCMG, while the thickness of pole pieces and magnets is the most effective factor in the designing procedure of TSCMG. Further, the final structure was designed and optimised using FEM and compared with the preliminary design. The comparison results showed that the flux density and rotor torque of TSCMG have been improved dramatically in the optimised configuration. Finally, to verify the proposed structure, a prototype of triple-speed gear was prepared. The experimental results showed a good agreement with the simulation results, with a maximum error of 4%-7% in the inner, middle and outer rotor torques. According to Figures 16 and 17, the difference between experimental and simulation results is acceptable. The simulated and empirical data confirmed that TSCMG can be considered for a wide range of industries and will be a powerful gear in a special application, like permanent wind generation, that needs three separate speeds to increase the productivity of wind power generation. -413