Torque ripple investigation in coaxial magnetic gears

. Magnetic gears offer significant advantages such as low noise and vibration level, lower maintenance and higher reliability compared to mechanical gears and are suitable for many applications in the industry. The coaxial magnetic gear has been extensively discussed in the literature, since it achieves higher torque densities amongst other magnetic gear configurations. The magnetic field is generated by permanent magnets mounted on the two rotors and a modulator between them. The modulator consists of ferromagnetic segments that are typically encased in a resin in order to increase its stiffness without compromising the generated magnetic field. However, due to the development of radial forces, oscillations of the ferromagnetic segments occur, which lead to torque ripples that affect the operation of the coaxial magnetic gear drive in applications where accuracy is required. This work introduces a computationally lightweight analytical 2D model in order to determine the applied radial force on the ferromagnetic segments at each angle of rotation of the two rotors and henceforth calculate the displacement of these segments using FEA. In this way it is possible to assess the variation of the torque (ripple) versus the angle of rotation of the input or output shaft. A parametric investigation examining the influence of the ferromagnetic segment thickness on the resulting torque ripple of a specific drive was carried out illustrating the benefits of the analytical models developed herein.


Introduction
Mechanical gears have been extensively used in power transmission applications due to the high torque densities they can achieve. However, mechanical gears suffer from some inherent problems such as noise, friction, requirement for lubrication, wear, fatigue and as a consequence poor reliability. On the other hand, magnetic gears (MGs) have a significantly lower level of noise, vibration and wear, therefore require a much lower maintenance cost compared to mechanical gears. In addition, MGs have overload protection since there is no contact between the shafts, leading to higher reliability [1]. As a consequence, MGs are suitable for a variety of applications [2] such as in aircraft mechanical transmission [3][4][5], wind power generation [6][7][8][9], wave energy conversion [10], traction [11] and aerospace [12]. The idea of MGs was first established in the early 20 th century. Since their introduction a variety of topologies have been proposed by researchers however, the low torque densities and in general their poor performance were limiting factors in their wide application in the industry [13,14]. In 2001, Atallah et al. [15] proposed a novel topology named coaxial MG (CMG) that could achieve high torque densities, due to the fact that during operation all the permanent magnets (PMs) contribute simultaneously in the generated magnetic field. The CMG consists of two concentric iron yokes, the PMs that are mounted on them and a flux-modulator ring that is placed between them. The modulator ring consists of ferromagnetic segments that are typically encased in a resin in order to increase its stiffness without compromising the generated magnetic field [16]. The number of ferromagnetic segments used in the modulator ring is equal to the sum of the pole pairs of the PMs in the inner rotor and the pole pairs of the PMs in the outer rotor. Therefore, from the generated magnetic field of the CMG drive torques are applied in the two rotors.
Torque ripple is a phenomenon that requires thorough investigation in power transmission systems since it could affect applications where accuracy is required. Torque ripple appear during the operation of the CMG drive since the torque in the two rotors is a sum of infinite sinusoidal harmonic terms. The torque ripple caused from the harmonic terms has significant impact in low pole-pair number configurations of CMG drives, while its effect on high pole-pair number configurations is usually negligible [16]. In addition, during the operation of the CMG drive radial forces are produced and applied to the ferromagnetic segments due to the magnetic field generated from the PMs in the inner and outer rotor. As a consequence, oscillations of the ferromagnetic segments occur, which lead to torque ripple that could affect the operation of the CMG drive. Therefore, a computationally lightweight analytical 2D model that will determine the applied radial forces on the ferromagnetic segments and the consequent torque ripple will be very beneficial to both researchers and the industry.
In the present work, the torque ripple in CMG drives caused by the displacement of the ferromagnetic segments will be determined. The magnetic flux density in the radial and tangential direction of the CMG drive can be calculated analytically [17] and therefore both the torque and radial forces can be obtained at every angle of rotation of the two rotors though the Maxwell Stress Tensor [18][19][20][21]. The displacement of the ferromagnetic segments caused by the radial forces will be obtained through FEA. Therefore, for the new position of the ferromagnetic segments at the following time step, the torque can be calculated and consequently the variation of torque in the two rotors can be determined. To illustrate the above phenomenon a case study will be performed where the inner rotor will be rotated and the outer rotor will be stationary, therefore the torque ripple due to the oscillation of the modulator ring will be calculated. Furthermore, the influence of the thickness of the ferromagnetic segments in the torque ripple of the CMG drive will be investigated.

Magnetic Potential Calculation
The Coaxial Magnetic Gear (CMG) consists of three parts: the inner rotor, the outer rotor and the flux modulator ring. As shown in Fig. 1, 1 , 2 , 3 , 4 , 5 , 6 , are the radii of the inner iron yoke, the inner PMs, the inner modulator ring's side, the outer modulator ring's side, the outer PMs, the outer iron yoke and the outer side of the CMG respectively. In addition, and are the right and left border of the ℎ ferromagnetic segment. Jian et al. [17] developed an analytical model for the calculation of the scalar magnetic potential. The total magnetic field created by the permanent magnets (PMs) can be calculated as a superposition of the magnetic fields created by the magnets of each rotor separately. Therefore, two models are constructed: one without the outer rotor's PMs and one without the inner rotor's PMs.
For the analytical calculation of the magnetic field in each respective model, the assumption of infinite permeability of the iron yokes and the ferromagnetic segments is considered.
The fundamental equations used to develop the analytical model are the Maxwell's equations (Gauss and Ampere Law): where B is the magnetic flux density vector and H is the magnetic field density vector. As a consequence, H can be written in the following form: where is the scalar magnetic potential.

Fig. 2.
Regions of the analytical model.
In Fig. 2, the first model -without the outer rotor's PMs-is presented. Region I represents the PMs of the inner rotor, Region II is the airgap below the modulator ring and Region III is the space between the above side of the modulator ring and the outer rotor's back iron. The relation of the magnetic flux density vector and the magnetic field density vector, in each region can be written as: where 0 is the vacuum permeability, is the relative permeability of the PMs and is the residual magnetization vector of the PMs.
From Eq. (1)-(5) the partial differential equation (PDE) can be obtained for each region: The general solution of Eq. (7) for , respectively is: The general solution of Eq. (6), is obtained as a superposition of the general solution of the Laplace Equation (Eq. (7)) and a special solution. In order to obtain the special solution, the magnetization distribution has to be expressed in an analytical and continuous function with the application of Fourier Series method.
In order to obtain the general solution in the slots, the magnetic potential has to be described in an analytical and continuous form with the use of Fourier series. The magnetic potential at the modulator ring, at the boundary radius is: The general solution of the Eq. (8) is obtained: where is the magnetic potential of the ℎ ferromagnetic segment. In order to determine the coefficients of Eq. (9), (10), (12), (13), certain boundary conditions are applied [25]: 1. Zero magnetic potential in radii 1 and 6 ( | 1 = 0, | 6 = 0). 2. Continuity of the magnetic potential and continuity of its radial derivative along adjacent regions. 3. Continuity of the flux density through the slots at 3 and 4 . 4. The magnetic flux flowing through the inside surface should be equal to the flux at flowing through the outside surface. 5. The flux flowing into each ferromagnetic segment should be equal to the flux flowing out. From the boundary conditions a linear system of (2Nn+N+12n+6) equations is derived, where n is the number of solutions of the PDEs of the system, from which the unknown coefficients of the magnetic flux density can be obtained. Therefore, the radial and tangential magnetic flux is: Following the same methodology, the respective coefficients of the second model -without the inner rotor's PMscan be calculated.

Torque and radial forces
For the calculation of the torque in the inner and outer rotor the Maxwell Stress Tensor [18][19][20][21] is applied. The torque in the inner rotor can be obtained, after the superposition of the magnetic flux density fields generated from the inner and outer rotor's PMs, as follows: The torque in the outer rotor can be obtained from: The radial force applied in the modulator ring can be obtained from: The force applied to the slots is equal to zero. Therefore, the radial force applied at each ferromagnetic segment is: Both torques in the two rotors and the radial force applied can be calculated numerically since both the radial and tangential flux density are obtained analytically.
The applied force at each ferromagnetic segment will result to its displacement leading to a different relative distance between the ferromagnetic segments and the inner and outer rotor's PMs. Therefore, for the different geometry at each time step the magnetic field will fluctuate and therefore the resulting torques in the two rotors will fluctuate as well resulting to torque ripple during the operation of the CMG drive. It should be noted that the displacement of each ferromagnetic segment is different along the z-axis since it is supported in one end and free at the other. Therefore, the displacement of each ferromagnetic segment will be similar to a cantilever beam with additional support due to the resin in the slots. As a consequence, along the zaxis the relative distance between the ferromagnetic segments and the inner and outer rotor's PMs will be different. Therefore, the applied torque at the rotors can be obtained numerically along the z-axis as follows: where and are the different torque values appear along the z-axis at the inner and outer rotor respectively.

Results and discussion
For the investigation of the torque ripple phenomenon due to the oscillations of the modulator ring a case study will be performed where the inner rotor will rotate while the outer rotor will be held stationary. The applied torque, at each angle of rotation, in the inner rotor will be calculated initially without taking into consideration the oscillations of the modulator ring. Then the applied torque will be calculated including the displacement of the ferromagnetic segments caused by the radial forces. The deviation between the two calculated torques will yield the equivalent torque ripple. The geometric parameters of the CMG drive are given on Table 1. The parameter of investigation was the radial thickness of the modulator ring and it is expressed as a percentage of the air space between the two rotors. For the displacement calculation at the modulator ring, the radial forces derived from (17.A) were used to implement a Finite Element Analysis. A 3D model of the modulator ring (Fig. 4) was constructed with the ferromagnetic segments supported by epoxy resin to increase the stiffness of the modulator ring. The mesh type used was tetrahedral and a bonding boundary condition was applied between the ferromagnetic segments and the epoxy resin segments. The results of the FE simulation for percentage equal to 65% are presented in Fig. 5.   Fig. 5. FE simulation to calculate the radial displacement of the modulator ring.
The torque ripple generated at the inner rotor for every angle of the inner rotor's rotation is presented in Fig. 6. It can be observed that the maximum torque ripple due to the oscillation of the modulator ring is about 0.62%. Finally, the effect of the percentage is presented in Fig. 7 where it can be observed that the maximum value of the torque ripple is reduced when the radial thickness of the modulator ring is increased.

Conclusions
In the present research the torque ripple due to the oscillations of the modulator ring on the induced torque of CMG drives was investigated. During the operation of CMG drives radial forces are generated by the magnetic fields of the PMs of the two rotors resulting to the displacement of the ferromagnetic segments of the modulator ring. The oscillations of the modulator ring will alter the geometry of the CMG drive and consequently introduce torque ripple during its operation. For a given configuration of a CMG drive and angle of rotation of the two rotors the radial and tangential magnetic flux density is calculated analytically and consequently the induced torque and the radial forces can be determined from the Maxwell Stress Tensor numerically. Therefore, the displacement of the modulator ring can be determined from a FE analysis and consequently, for the new geometry the torque can be determined implementing the same process. A case study was performed where the inner rotor was rotated and the outer rotor was held stationary. The induced torque in the inner rotor was initially calculated without taking into account the oscillation of the modulator ring. Then, the induced torque was calculated with the inclusion of the displacement of the ferromagnetic segments. The difference between the values of the two torques at each respective angle of rotation yielded the equivalent torque ripple. Furthermore, the effect of the radial thickness of the modulator ring, as a percentage of the air space between the two rotors, on the torque ripple was investigated. The maximum value of the torque ripple reduced as the percentage of the radial thickness increased. However, it was observed that the maximum value of the torque ripple was over 0.5% in all cases therefore the effect of the oscillation of the modulator ring during the operation of CMG drives could be significant in applications where high accuracy is required.