Fast Analysis and Optimization of a Magnetic Gear Based on Subdomain Modeling

Abstract

This study presents a two-dimensional analytical method for fast optimization, taking into consideration the influence of the eddy current in a magnet and iron loss within a coaxial magnetic gear. Subdomain modeling was utilized to obtain vector potentials in the air-gap, magnet, and modulation regions by solving Maxwell’s equations. After that, the magnet, rotor, and modulation losses were predicted and then compared using a finite element method simulation within three topologies with gear ratios ranging from five to six. The authors improved the machine performance, specifically the torque density, by employing a multi-objective function with particle swarm optimization. The flux density obtained using subdomain modeling in just 0.5 s benefits the optimization process, resulting in a torque-density optimal model after around 3 h. A 3/19/16 prototype targeting a low-speed, high-torque, permanent generator application was fabricated to verify the analytical and simulation results.

1. Introduction

In contrast to traditional mechanical gears, magnetic gears (MGs) transmit power between high-torque, low-speed rotations and low-torque, high-speed rotations through the interaction of magnetic fields rather than physical contact between interlocking teeth. The potential benefits of magnetic gears include inherent overload protection, decreased maintenance requirements, enhanced reliability, and physical isolation between shafts. Consequently, magnetic gears can be promising candidates for use in gearless or direct-drive magnetically geared machines (MGMs) due to their advantageous power density. The concept of the coaxial magnetic gear was initially introduced in 1968 [1], with its operational principles and potential further explained in previous work [2,3].

Over the last twenty years, many studies have been conducted to investigate various power, topology, and application ranges. A comprehensive overview of the evolution and numerous approaches to MGMs has also been published [4,5,6]. Prototypes of magnetic gears have been investigated for many other potential uses, such as aircraft propulsion (NASA) [7,8], wind [9,10,11,12], wave [13,14,15], marine [16], traction [17,18,19,20,21], and space [22,23,24] applications. The torque density of MGs was reviewed and compared in a previous study [25]. Three typical examples published in [10,26] and [27] achieved torque densities of 110 $\mathrm{k}\mathrm{N}\mathrm{m}/{\mathrm{m}}^3$, 141 $\mathrm{k}\mathrm{N}\mathrm{m}/{\mathrm{m}}^3$, and 239 $\mathrm{k}\mathrm{N}\mathrm{m}/{\mathrm{m}}^3$, respectively. In addition to coaxial radial flux MGs, some papers have also paid attention to other structures: the work in [28,29,30,31] investigated axial flux counterparts; however, the performance of the axial analog was much lower than the conventionally radial counterparts [31]. Consequently, linear topologies, which are the best candidates for wave generators, were introduced [15,32,33,34,35]. Prior work has presented [36,37] and minimized [2,38,39] cogging torque by strategically selecting the number of pole pairs, unequal-space-type pole pieces, and skewing.

When analyzing electrical machines in general or MGs in particular, the air-gap flux density is always the first and most essential quantity that designers concern themselves with. The aforementioned studies mainly employed FEM simulation, resulting in significant computational time or magnetic equivalent circuits, which do not predict accurately. The analytical method, on the other hand, offers faster running times and results that agree well with FEM simulations, making it a promising solution for MG pre-design and optimization. Two analytical methods are mainly studied. The first method (subdomain modeling) is based on directly solving Maxwell’s equations and the boundary and interface conditions by using the Fourier series expansion method. Researchers have applied the subdomain method to various cases, such as radial and axial magnetic gears, eccentricity effects in magnetic gears, and magnetically geared machines, as demonstrated in [40,41,42,43]. The second method (harmonic modeling), which solves permeability modeling using the complex Fourier series and Cauchy product, allows the consideration of nonlinear characteristics in the core [44,45]. Based on the analytical results, the authors of [43,46] optimized the output power using particle swarm optimization (PSO) and genetic algorithms (GAs). Dealing with mechanical analyses, the authors of [47] utilized 3D FEM, which required several hours to obtain results; the authors of [48] employed the subdomain method to predict stress and deformation in the modulation in seconds; and the authors of [49] experimentally verified a laminated structure in modulation after simulating a relevantly simplified model.

This paper extensively examines a coaxial MG, employing both subdomain and harmonic reduction methods to predict the air-gap flux density and enhance the computational efficiency. Following this approach, the core-loss and eddy-current effects were analyzed across different configurations, and the machine underwent optimization to boost torque density. Ultimately, a 3/16/19 prototype was manufactured, and the torque capacity results obtained from the FEM simulation and the proposed method were verified via the test bench.

2. Principle of Magnetic Gear

The magnetic gear shown in Figure 1a is composed of three parts: an inner rotor with $Z_{pi}$ pole pairs of the permanent magnet; an outer rotor with $Z_{po}$ pole pairs of the permanent magnet; and modulation with $P_m$ poles. The transmission of flux between the inner rotor and the outer rotor and vice versa requires a minimum of three pole numbers as follows [2]:

$$Z_{pi}+Z_{po}=P_m$$

(1)

If the modulation is fixed and machine losses are ignored, the gear ratio of the inner and outer rotors is given by [3]

$$G={\displaystyle \frac{{\omega }_i}{{\omega }_o}}={\displaystyle \frac{T_o}{T_i}}={\displaystyle \frac{Z_{po}}{Z_{pi}}}$$

(2)

where $T_o$ and $T_i$ are the electromagnetic torque of the external and internal rotors, respectively.

To reduce the cogging torque, selecting a magnet pole number that does not result in an integer gear ratio has been recommended [2], which is helpful when selecting a pole-pair combination. For instance, for a gear ratio requirement of six, three pole-pair combinations ($(Z_{pi}$/$Z_{po}$/$P_m$) = {(2/11/13); (3/16/19); (3/17/20)}) should be selected. In the current study, the authors also verified these combinations related to loss calculations, as discussed in Section 4.

3. Subdomain Modeling

Figure 1b shows the MG and simplified analytical models employed in the subdomain method. Before obtaining the solutions, the following assumptions were made:

• The end effects are neglected;
• The problem is 2D in cylindrical coordinates;
• The magnetic vector potential $A$, current density $J$, magnetization vector $M$, and magnetic flux density vector $B$ have the following non-zero components:

  $$A=[\mathrm{0,0},A_z]; J=[\mathrm{0,0},J_z]; M=[M_r,M_t,0]; B=[B_r,B_t,0];$$
• The core materials have infinite permeability;
• The shaft is a non-magnetic material.

3.1. Governing Partial Differential Equations (PDEs)

By conducting a 2D analysis in cylindrical coordinates, the partial differential equations (PDEs) for the five regions are expressed as follows:

$${\displaystyle \frac{{\partial }^2{A}_z^{I,IV}}{\partial r^2}}+{\displaystyle \frac{\partial A_z^{I,IV}}{r\partial r}}+{\displaystyle \frac{{\partial }^2{A}_z^{I,IV}}{r^2\partial {\theta }^2}}={\displaystyle \frac{{\mu }_0}{r}}\left(M_t^I+{\displaystyle \frac{\partial M_r^{I,IV}}{\partial \theta }}\right)$$

(3)

$${\displaystyle \frac{{\partial }^2{A}_z^{II,III}}{\partial r^2}}+{\displaystyle \frac{\partial A_z^{II,III}}{r\partial r}}+{\displaystyle \frac{{\partial }^2{A}_z^{II,III}}{r^2\partial {\theta }^2}}=0$$

(4)

$${\displaystyle \frac{{\partial }^2{A}_z^p}{\partial r^2}}+{\displaystyle \frac{\partial A_z^p}{r\partial r}}+{\displaystyle \frac{{\partial }^2{A}_z^p}{r^2\partial {\theta }^2}}=0$$

(5)

where $M_r$ and $M_t$ are the radial and tangential magnetizations of the parallel magnet, respectively, whose description can be found in [43,48]. The general PDE solution is the sum of the homogeneous and particular solutions. Owing to the magnetic excitation in the magnet regions, the solutions contain both homogeneous and particular solutions, whereas in the air-gap regions, homogeneous solutions alone are sufficient. The assumption of infinite permeability in the core leads to tangential-direction boundary conditions in the modulation region, ${\left.{\displaystyle \frac{\partial A_z^p}{\partial \theta }}\right|}_{\theta ={\theta }_p}={\left.{\displaystyle \frac{\partial A_z^p}{\partial \theta }}\right|}_{\theta ={\theta }_p+\alpha }$. Therefore, the vector potential of each region can be expressed as

$$A_z^{I,IV}=\sum _{n=\mathrm{1,2}}^{\infty }\begin{array}{c}\left(r^n{A}_n^{I,IV}+r^{-n}{B}_n^{I,IV}+P_n^s\right)\sin \left(n\theta \right)+\\ \left(r^n{C}_n^{I,IV}+r^{-n}{D}_n^{I,IV}+P_n^c\right)\cos \left(n\theta \right)\end{array}$$

(6)

$$A_z^{II,III}=\sum _{n=\mathrm{1,2}}^{\infty }\begin{array}{c}\left(r^n{A}_n^{II,III}+r^{-n}{B}_n^{II,III}\right)\sin \left(n\theta \right)+\\ \left(r^n{C}_n^{II,III}+r^{-n}{D}_n^{II,III}\right)\cos \left(n\theta \right)\end{array}$$

(7)

$$A_z^p=A_0^p+\ln \left(r\right)B_0^p+\sum _{h=\mathrm{1,2}}^{\infty }\left(r^{h{\displaystyle \frac{\pi }{\alpha }}}{C}_h^p+r^{-h{\displaystyle \frac{\pi }{\alpha }}}{D}_h^p\right)\cos \left(h{\displaystyle \frac{\pi }{\alpha }}\left(\theta -{\theta }_p\right)\right)$$

(8)

where $P_n^s$ and $P_n^c$ were presented in [48].

The continuity of the vector potential radial components leads to the following boundary conditions:

$$r=R_6\to {\displaystyle \frac{\partial A_z^I}{{\mu }_r\partial r}}+{\mu }_0{M}_t^I=0$$

(9)

$$r=R_5\to \left\{\begin{array}{c}{\displaystyle \frac{\partial A_z^I}{{\mu }_r\partial r}}+{\mu }_0{M}_t^I={\displaystyle \frac{\partial A_z^{II}}{\partial r}}\\ A_z^{II}=A_z^I\end{array}\right.$$

(10)

$$r=R_4\to \left\{\begin{array}{c}{\displaystyle \frac{\partial A_z^{II}}{\partial r}}={\displaystyle \sum _{p=\mathrm{1,2}}^{P_m}}{\displaystyle \frac{\partial A_z^p}{\partial r}}\\ A_z^p=A_z^{II}\end{array}\right.$$

(11)

$$r=R_3\to \left\{\begin{array}{c}{A}_z^p=A_z^{III}\\ {\displaystyle \frac{\partial A_z^{III}}{\partial r}}={\displaystyle \sum _{p=\mathrm{1,2}}^{P_m}}{\displaystyle \frac{\partial A_z^p}{\partial r}}\end{array}\right.$$

(12)

$$r=R_2\to \left\{\begin{array}{c}{A}_z^{III}=A_z^{IV}\\ {\displaystyle \frac{\partial A_z^{IV}}{{\mu }_r\partial r}}+{\mu }_0{M}_t^{IV}={\displaystyle \frac{\partial A_z^{III}}{\partial r}}\end{array}\right.$$

(13)

$$r=R_1\to {\displaystyle \frac{\partial A_z^{IV}}{{\mu }_r\partial r}}+{\mu }_0{M}_t^{IV}=0$$

(14)

3.2. Flux Density and Improving Computational Time

The flux density at the air gap and magnet are deduced from

$$B_r^{I,IV}={\displaystyle \frac{\partial A_z^{I,IV}}{r\partial \theta }}; B_t^{I,IV}=-{\displaystyle \frac{\partial A_z^{I,IV}}{\partial r}}$$

(15)

In total, there are 20 coefficients in (6)–(8). If the harmonic number is $N$ = 150 and $K$ = 3, the matrix size is $16N+4P_{m}K$ × $16N+4P_{m}K$ (2628 × 2628). Conversely, a small matrix size results in an inaccurate solution. However, the larger the size, the greater the computational time. In this study, the harmonic number $N$ was first reduced; then, the proposed approach was employed, as discussed in the following sections.

In [50,51], the harmonic content of the machine was categorized into two groups: the source term, including the magnets, and the spatial aspects, which encompass the machine’s geometry and winding distribution. Understanding the machine structure can help to predict the harmonics that contribute to the output parameters. Therefore, non-useful harmonics can be detected.

Subsequently, a 3/16/19 specification, whose parameters are listed in

Table 1. Specification parameters.

Quantity	Symbol	Unit	Value
Outer rotor radius	$R_7$	mm	39.0
Outer external magnet radius	$R_6$	mm	32.2
Inner external magnet radius	$R_5$	mm	30.7
Outer modulation radius	$R_4$	mm	29.7
Inner modulation radius	$R_3$	mm	24.7
Outer internal magnet radius	$R_2$	mm	23.7
Inner internal magnet radius	$R_1$	mm	19.5
Inner rotor radius	$R_0$	mm	10.0
Stack length	$L_{stk}$	mm	77.0
Remanent of magnets	$B_0$	T	1.25
Magnet relative permeability	${\mu }_r$	-	1.03
Vacuum permeability	${\mu }_0$	$\mathrm{k}\mathrm{g}·\mathrm{m}·{\mathrm{s}}^{-2}·{\mathrm{A}}^{-2}$	$4\times {10}^{-7}$
Outer magnet pole pair	$Z_{po}$	-	11/16/17
Inner magnet pole pair	$Z_{pi}$	-	2/3/3
Modulation pole	$P_m$	-	13/19/20
Outer magnet pitch ratio	$\alpha$	-	1.0
Inner magnet pitch ratio	$\beta$	-	1.0
Modulation pitch	$\gamma$	$2\mathsf{\pi }/{\mathrm{P}}_{\mathrm{m}}$ rad	0.5

, is computed. As shown in Figure 2, there is good agreement between the subdomain method and FEM simulation results in terms of flux density. Additionally, the radial and tangential flux densities were analyzed via the fast Fourier transform (FFT) to evaluate the contribution of the harmonic order. Notably, the FFT analysis results for the flux densities at the air gap and magnet show that not all harmonics contribute to the total flux density; however, they consume more computational time. For the flux density derivation, if only dominant harmonics having an amplitude of over 2% of the maximum amplitude remain, the computational time is reduced from 1.6 to 0.5 s without effecting the accuracy. This time reduction benefits the optimization thereafter.

4. Loss Definition

The loss in MGs is mainly composed of three components: modulation loss, rotor yoke loss, and eddy-current loss in the magnet. During operation, the modulation is stationary; therefore, its flux flow behavior is identical to that of the stator teeth of synchronous machines. The rotor yoke loss in conventional machines is often neglected owing to non-frequency-generating synchronous rotation. Conversely, the fluxes flowing through the rotors in the MG are coupled to each other. This interconnection causes more than one frequency in each rotor yoke, thereby generating a significant loss. This can be seen clearly in the rotor yoke flux density shown in Figure 3, in which the flux of the three-pole-pair inner magnet flows through the modulation, and then interacts with the outer magnet and results in third-order harmonic occurrence. The magnets exhibited a similar phenomenon, which significantly influenced the eddy-current effect. The detailed implementation is discussed in the following sections.

4.1. Modulation Loss

The core loss can generally be estimated as the sum of the hysteresis, eddy-current, and excess losses [52]. Notably, the flux density variation in the machine is not purely sinusoidal. Therefore, the harmonic effects should be reflected to predict the eddy-current and excess losses, as follows:

$$P_{core}=P_h+P_c+P_e=V_{core}\left(C_{dc}{k}_{h}f{B}_m^2+\sum _{v=1}^{odd}{k}_c{\left(fv\right)}^2{B_{mv}}^2+k_e{\left(fv\right)}^{{\displaystyle \frac{3}{2}}}{B_{mv}}^{{\displaystyle \frac{3}{2}}}\right)$$

(16)

where $V_{core}$ is the volume. The coefficients $k_h$, $k_c$, and $k_e$ are the hysteresis, eddy current, and excess factors, respectively, which can be derived using the curve-fitting technique and loss data provided by the manufacturer [53,54]. In this study, the coefficients employed in the FEM and calculations were defined as $k_h=386$, $k_c=0.061$, and $k_e=15.8$. $f$ is the electrical frequency of modulation flux, which is calculated the same way as that of a synchronous machine. $B_m$ is the peak flux density. $v$ and $B_{mv}$ are the harmonic order and the corresponding flux density, respectively, obtained via FFT analysis with respect to time. Finally, $C_{dc}$ represents the bias effect of the DC-component flux density and is defined as $C_{dc}=1+0.65B_{dc}^2$. In this study, the modulation characterized the AC magnetic field; therefore, $B_{dc}=0$. Additionally, alternating and rotating fields based on the time harmonic loci were introduced [45,46,53,54,55,56]. However, this effect was not considered in this study.

Because of the infinite permeability assumption at the start, the subdomain method does not allow direct flux density prediction at modulation. A secondary approach using the flux linkage results for two adjacent slots was employed to compute the maximum value. The flux linkage and density at point A in Figure 3a can be expressed as follows:

$${\Phi }_p={\displaystyle \frac{L_{stk}}{\frac{R_4^2-R_3^2}{2}\gamma }}\underset{R_3;{\theta }_p}{\overset{R_4;{\theta }_p+\gamma }{\iint }}{A}_r^p(\theta ,z)rdrd\theta$$

(17)

$$B_p^m={\displaystyle \frac{{\Phi }_p+{\Phi }_{p+1}}{2}}{\displaystyle \frac{1}{L_{stk}\frac{R_4+R_3}{2}\gamma }}$$

(18)

A new value is obtained for each new rotor position. After one electrical period, an array containing the flux density values is achieved. In (16), the maximum value in the array can be used as $B_m$, whereas all array elements are analyzed by FFT to deduce the harmonics and their amplitudes, which are denoted as $v$ and $B_{mv}$, respectively.

4.2. Rotor Yoke Loss

The rotor yoke parts are assumed to have infinite permeability. Therefore, flux density, in particular core loss, is predicted through an equivalent transformation. First, the flux densities at radii $R_1$ and $R_6$ are defined by the vector potentials of regions I and IV, as shown in Figure 1b.

$$B_r^{I\left(IV\right)}={\left.{\displaystyle \frac{\partial A_z^{I\left(IV\right)}}{r\partial \theta }}\right|}_{r=R_6 \left(R_1\right)}; B_t^{I\left(IV\right)}={\left.-{\displaystyle \frac{\partial A_z^{I\left(IV\right)}}{\partial r}}\right|}_{r=R_6 \left(R_1\right)}$$

(19)

Similar to the flux densities of the external and internal magnets in Figure 2a,d, the flux densities are composed of many harmonics. These flux densities, which are considered to be the two largest components, can be estimated as the sum of the $Z_{po}^{th}$- and $Z_{pi}^{th}$-order harmonics, as shown in Figure 3. Owing to the synchronous speed of each rotor, one harmonic does not generate a frequency; consequently, the corresponding order harmonic is eliminated. For example, in an external rotor with pole-pair number $Z_{po}$, the $Z_{po}^{th}$-order harmonic is neglected in the loss calculation. In an internal rotor with pole-pair number $Z_{pi}$, the $Z_{pi}^{th}$-order harmonic is removed. The frequencies of both rotors match the frequencies of the magnets, which are expressed as

$$f_{I\left(IV\right)}=f{\displaystyle \frac{P_m}{Z_{po\left(i\right)}}}$$

(20)

The maximum rotor yoke flux density can be obtained by

$$B_{r\left(t\right)\_ry}^I=B_{r\left(t\right)}^{I(Z_{pi}^{th})}{\displaystyle \frac{\pi R_7}{Z_{pi}}}{\displaystyle \frac{2}{\pi }}{\displaystyle \frac{1}{2}} {\displaystyle \frac{1}{R_7-R_0}}; B_{r\left(t\right)\_ry}^{IV}=B_{r\left(t\right)}^{IV\left(Z_{po}^{th}\right)}{\displaystyle \frac{\pi R_1}{Z_{po}}}{\displaystyle \frac{2}{\pi }}{\displaystyle \frac{1}{2}} {\displaystyle \frac{1}{R_1-R_0}}$$

(21)

The first equation in (21) can be explained in more detail as follows: $B_{r\left(t\right)}^{I\left(Z_{pi}^{th}\right)}$ is the radial (or tangential) flux density in the external rotor yoke, which remains in the $Z_{pi}^{th}$ order after the FFT analysis. Supposing that all fluxes traverse through the rotor yoke, ${\displaystyle \frac{\pi R_7}{Z_{po}}}{\displaystyle \frac{2}{\pi }}$ is the average flux linkage calculation. This flux flows equally to the two sides of the yoke and is represented by ${\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{R_8-R_7}}$.

The flux densities in (16) are replaced by the radial and tangential flux densities in (21), and the core loss is expressed as

$$P_{core}=V_{core}\left(\begin{array}{c}{C}_{dc}{k}_h{f}_{I\left(IV\right)}\left({\left(B_{r_{ry}}^{I\left(IV\right)}\right)}^2+{\left(B_{t_{ry}}^{I\left(IV\right)}\right)}^2\right)+\\ k_c{f}_{I\left(IV\right)}^2\left({\left(B_{r_{ry}}^{I\left(IV\right)}\right)}^2+{\left(B_{t_{ry}}^{I\left(IV\right)}\right)}^2\right)+\\ k_e{f}_{I\left(IV\right)}^{{\displaystyle \frac{3}{2}}}\left({\left(B_{r_{ry}}^{I\left(IV\right)}\right)}^{{\displaystyle \frac{3}{2}}}+{\left(B_{t_{ry}}^{I\left(IV\right)}\right)}^{{\displaystyle \frac{3}{2}}}\right)\end{array}\right)$$

(22)

The coefficient $C_{dc}$ is defined in a manner similar to the modulation loss calculation. However, in the rotor yoke, the flux density consists of both AC and DC terms; therefore, $B_{dc}$ should be defined.

4.3. Magnet Loss

As mentioned previously, the coupling effect between the magnet numbers of the two rotors generates a frequency in the magnet. In this machine, the magnet layers are modeled in regions I and IV; therefore, the flux density and frequency can be calculated directly. First, the frequencies of the inner and outer magnets are given by

$$f_{mi\left(o\right)}=f{\displaystyle \frac{P_m}{Z_{pi\left(o\right)}}}$$

(23)

When the rotors rotate, the flux density at the middle points of the magnet (points B and C in Figure 3a) is also analyzed using FFT to deduce the harmonic order and their amplitudes, which were expressed as $v$ and $B_{mv}^{PM}$, respectively. Notably, the components related to the magnetization in (6) do not contribute to the eddy-current loss; therefore, they can be removed from the flux density calculation in the magnet loss section.

$$A_{PM}^{I,IV}=\sum _{n=\mathrm{1,2}}^{\infty }\begin{array}{c}\left(r^n{A}_n^{I,IV}+r^{-n}{B}_n^{I,IV}\right)\sin \left(n\theta \right)+\\ \left(r^n{C}_n^{I,IV}+r^{-n}{D}_n^{I,IV}\right)\cos \left(n\theta \right)\end{array}$$

(24)

Then, calculation of the eddy current in the magnets can be approached as shown in Figure 3.

$$J_{PM}=-{\sigma }_{PM}{\displaystyle \frac{\partial A_{PM}}{\partial t}}+C(t)$$

(25)

where ${\sigma }_{PM}$ is the conductivity of the magnet and $C\left(t\right)$ is a term ensuring the total current zero within every single magnet. In the coordinate $Or\theta$, $C\left(t\right)$ can be expressed by

$$C\left(t\right)={\displaystyle \frac{2}{{\theta }_{PM}}}{\displaystyle \frac{1}{r_o^2-r_i^2}}\underset{r_i}{\overset{r_o}{\int }}\underset{{\theta }_j}{\overset{{\theta }_j+{\theta }_{PM}}{\int }}{\sigma }_{PM}{\displaystyle \frac{\partial A_{PM}}{\partial t}}rdrd\theta$$

(26)

Figure 3b shows the flux density vector and vector potential in a magnet piece. It can be seen that the vector potential is symmetrical along the z-axis, which describes the zero-current characteristic in the magnet piece. By re-choosing a new coordinate $O{r}^{\prime }\theta$, the integral equation of $C\left(t\right)$ can be rewritten as follows:

$$C\left(t\right)={\displaystyle \frac{2}{{\theta }_{PM}}}{\displaystyle \frac{1}{r_o^2-r_i^2}}\underset{r_i}{\overset{r_o}{\int }}\underset{-{\displaystyle \frac{{\theta }_{PM}}{2}}}{\overset{{\displaystyle \frac{{\theta }_{PM}}{2}}}{\int }}{\sigma }_{PM}{\displaystyle \frac{\partial A_{PM}}{\partial t}}rdrd\theta$$

(27)

In this new coordinate, $A_{PM}\left(\theta \right)=-A_{PM}\left(-\theta \right)$; $C\left(t\right)$, therefore, equals zero. Consequently, the eddy-current loss in the magnet segment is expressed simply as

$$P_{eddy}={\displaystyle \frac{L_{stk}}{{\sigma }_{PM}}}\underset{r_i}{\overset{r_o}{\int }}\underset{-{\displaystyle \frac{{\theta }_{PM}}{2}}}{\overset{{\displaystyle \frac{{\theta }_{PM}}{2}}}{\int }}{\left({\sigma }_{PM}{\displaystyle \frac{\partial A_{PM}}{\partial t}}\right)}^{2}rdrd\theta$$

(28)

Assuming that the tangential flux density does not contribute to the eddy-current effect, the vector potential in (28) can be approximated as $\partial A_{PM}=r\theta \partial \sum v{B}_{mv}^{PM}$. Finally, the eddy-current loss is given by

$$P_{eddy}=L_{stk}{\sigma }_{PM}\underset{r_i}{\overset{r_o}{\int }}\underset{-{\displaystyle \frac{{\theta }_{PM}}{2}}}{\overset{{\displaystyle \frac{{\theta }_{PM}}{2}}}{\int }}{\left({\displaystyle \frac{\partial }{\partial t}}\sum v{B}_{mv}^{PM}\right)}^2{r}^{3}dr{\theta }^{2}d\theta =L_{stk}{\sigma }_{PM}{\displaystyle \frac{r_o^4-r_i^4}{4}}{\displaystyle \frac{{\theta }_{PM}^3}{12}}\sum {\displaystyle \frac{{\left(2\pi f_{mo\left(i\right)}v{B}_{mv}^{PM}\right)}^2}{2}}$$

(29)

4.4. Loss Verification via FEM

Simulations were performed to validate the MG core loss. Three topologies characterizing the low cogging torque [2], whose parameters are listed in Table 1, were verified for various cases, as shown in Figure 4. The losses as a function of the rotor speed are shown in Figure 5. In general, the analytical results show good agreement with the FEM results, even though in high-speed operation resulting in a high electrical frequency the loss calculation error is slightly larger. The possible reasons are that (i) the estimated coefficients in the curve fitting step at high frequencies have a larger error than the low-frequency ones, and (ii) the loss depends exponentially on the frequency; thus, higher frequencies lead to a higher error possibility. Consequently, the influence of machine parameters on the loss can be summarized as follows:

• Compared to the full-pitch magnet in Figure 5a,c,e, decreasing the magnet pitch in the rotors in Figure 5b,d,f reduces the losses. This can be attributed to the fact that a smaller magnet pitch results in a smaller flux density.
• The loss of the small pole-pair combination $[Z_{po};Z_{pi}]$ = [11; 2] is significantly larger than that of groups [16; 3] and [17; 3] along all modulation frequency ranges. This is due to the contribution of the rotor yoke and magnet losses to the total loss. In (22) and (29), the losses depend on the rotor frequency, which is governed by the electrical frequency and pole-pair numbers in (20) and (23), respectively. Accordingly, as the number of pole pairs in the rotor decreases, the rotor frequency increases. This difference does not arise from modulation loss, as it can be attributed to the similarity of the electrical and modulation frequencies in (16).

It is worth noting that at the same primary speed, the modulation frequency of the [11; 2] pair is much lower than that of the other pairs. Consequently, a significant loss reduction can be achieved in this scenario. Therefore, the results shown in Figure 5 should not be used as a reference when selecting a particular specification. Determining the optimal choice requires a more comprehensive understanding of machine features such as cogging torque, power density, and efficiency.

5. Optimization Process

5.1. Variables, Objective Functions, and Constraints

After ensuring that the proposed mathematical model was accurate, an optimal solution of the 3/16/19 specification, whose parameters and optimal variables are listed in Table 1 and

Table 2. Value range of optimization.

Quantity	Symbol	Unit	Value
Outer magnet pitch ratio	$x_1=\alpha$	-	0.7–1.0
Inner magnet pitch ratio	$x_2=\beta$	-	0.7–1.0
Modulation pitch	$x_3=\gamma$	$2\mathsf{\pi }/{\mathrm{P}}_{\mathrm{m}}$ rad	0.4–0.6
Outer diameter	$x_4=R_7$	mm	23–100
Outer magnet width	$x_5=R_6-R_5$	mm	0.75–3
Modulation width	$x_6=R_4-R_3$	mm	2.5–10.0
Inner magnet width	$x_7=R_2-R_1$	mm	2.0–8.0

, respectively, was investigated in terms of efficiency and torque density. The objective function, equality function, and inequality constraints are expressed as

$$\begin{gathered} \underset{X=[x_1,\dots , x_7]}{\max }f\left(X\right)=\left[{\displaystyle \frac{P_e}{\pi R_8^2{l}_{stk}}}; {\displaystyle \frac{P_e}{P_e+Loss}}\right] \\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o} \\ 55<\mathrm{M}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{t} \mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e} \left[{\mathrm{d}\mathrm{m}}^3\right]<65 \\ {R}_7-R_6={\displaystyle \frac{\pi \alpha }{Z_{po}}}; R_1-R_0={\displaystyle \frac{\pi \beta }{Z_{pi}}} \end{gathered}$$

(30)

To prevent saturation, the rotor yoke thickness is set to the corresponding magnet pitch width. In PSO, a collection of particles representing potential solutions iteratively adjust their positions based on local and global information. Each particle moves toward its own best-known position and the overall best-known position within the swarm. Through iterations, the swarm gradually converges toward an optimal solution.

5.2. Optimization Results

The optimal flowchart employed in this study is illustrated in [43]. The swarm population size in one iteration and the maximum number of iterations were set to 15 and 20, respectively. Consequently, it took approximately 3 h to obtain an optimal specification, the dimensions of which are listed in

Table 3. Dimension comparison between the initial and optimal models.

Quantity	Symbol	Unit	Initial	Optimal
Outer rotor radius	$R_7$	mm	39.0	53.0
Outer external magnet radius	$R_6$	mm	32.2	46.0
Inner external magnet radius	$R_5$	mm	30.7	43.0
Outer modulation radius	$R_4$	mm	29.7	42.0
Inner modulation radius	$R_3$	mm	24.7	37.0
Outer internal magnet radius	$R_2$	mm	23.7	36.0
Inner internal magnet radius	$R_1$	mm	19.5	28.0
Inner rotor radius	$R_0$	mm	10.0	18.7
Stack length	$L_{stk}$	mm	77.0	25.6
Outer magnet pitch ratio	$\alpha$	-	1.00	0.84
Inner magnet pitch ratio	$\beta$	-	1.00	1.00
Modulation pitch	$\gamma$	$2\mathsf{\pi }/{\mathrm{P}}_{\mathrm{m}}$ rad	0.50	0.53

, and a Pareto curve is shown in Figure 6. The two Pareto fronts in Figure 6a are observed on the $yz$ and $xz$ planes in Figure 6b and Figure 6c, respectively. The dots represent all particles generated in the program. The red dots denote the points of maximum torque density when the magnetic field volume was varied. The efficiency was calculated as shown in Figure 6d. In general, multi-objective optimization is a compromise between two quantities. As can be seen, the maximum torque density and efficiency points do not occur synchronously. An increasing efficiency can result in a reduction in torque density and vice versa. Based on these requirements, the most suitable point was selected. For instance, in Figure 6, we chose and verified a green-dot point, whose structure is compared in Figure 7. The optimal solution has a larger diameter and shorter length than the initial solution.

Table 4. Optimization performance.

Characteristic	Initial Design		Optimal Design
	Subdomain	FEM	Subdomain	FEM
Torque—Nm	23.7	23.4	24.9	24.3
Torque density—kNm/m3	64	64	108	108
Magnet volume—dm3	66.1	66.1	54.9	59.4
Total loss—W	10.04	10.39	13.66	13.65
Efficiency—%	98.67	98.61	98.26	98.26

presents the optimization performance at an outer rotor speed of 300 rpm in terms of the torque, torque density, magnet volume, total loss, and efficiency obtained using the subdomain method and FEM. The optimal machine generated a higher output torque of approximately 5%, whereas it saved more magnetic resources by approximately 10%. Furthermore, the efficiency remained almost constant with a difference of <1%. Notably, these results show that the analytical and simulation results are in excellent agreement.

6. Experimental Validation

6.1. Three-Dimensional Simulation

The 3/16/19 specification is given in Table 1; however, the adjusted $R_7$ = 35 (mm) was simulated in 2D and 3D FEM, whose mesh settings are shown in Figure 8a and 8b, respectively. Meanwhile, the torque characteristics were calculated via subdomain modeling combined with the harmonic reduction method and then compared with the FEM results. As shown in Figure 9, there is a good agreement between the computation and simulation results. The 3D analysis gave an output torque that was approximately 7% lower than that of the subdomain method. Based on these results, the following conclusions can be drawn.

• The harmonic reduction approach result matches that of the original analytical approach while decreasing the running time.
• Saturation occurred because of the small rotor yoke thickness, which weakened the flux linkage. Therefore, in general, the FEM produced a slightly smaller torque.
• Three-dimensional FEM was taken into account by the endcap effect. Its torque was also reduced compared with the 2D results.
• The computational times of the 2D FEM, 3D FEM, and analytical method were 4 min, more than 60 min, and 25 s, respectively. This further highlights the subdomain modeling priority.

6.2. Comparison of Analysis and Experimental Results

In this study, a prototype was fabricated, as shown in Figure 10. The experimental process was as follows. The driving motor is the primary source of input torque in the outer rotor. Then, the MG transferred power from the outer side to the inner side. The MG output was used as the input power for the generator, which was connected to an adjustable load. Consequently, the generator and load system can be considered as the MG load. When the resistance value (generator load) or rotational speed (generator BEMF) fluctuates, the MG load also fluctuates. Owing to mechanical limitations, the outer-rotor speed range was restricted to 1000 rpm.

Figure 11a shows the outer rotor response when the inner rotor speed is varied. These results confirm the working principle of the MG. The rotor pole-pair combination was designed as 3/16 with an expected transformation ratio of 16/3 $\approx$ 5.33. All four points at 200, 400, 600, and 800 rpm show excellent agreement with the designed ratio.

To verify the maximum torque capacity of the MG, the load resistance was decreased, and the rotor speed was simultaneously increased to approximately 1000 rpm. This increase in speed was implemented slowly to ensure safety. The maximum torque was captured immediately as the rotor was synchronously lost. In Figure 11b, the captured maximum torque is 4.15 Nm at the 2-Ω resistance and 891-rpm speed point, which is close to the calculation (4.45 Nm). Notably, in the no-load results, the MG torque appeared to be relatively large, indicating an unpredictable impact of mechanical factors on the system. Therefore, experiments on loss and efficiency were not considered in this study.

7. Conclusions

The proposed subdomain approach combined with the harmonic reduction method was efficient in obtaining the magnetic gear characteristics. Loss calculations for the magnets, modulation, and rotor yokes were introduced based on the flux density in every region. Furthermore, compared with the 2D and 3D FEM simulations, the torque and loss computational time was reduced from dozens of minutes to roughly seven seconds, while maintaining excellent accuracy. This allows for the application of PSO techniques, which saves time exponentially. Consequently, the optimal model reached between 66 and 108 kNm/m³ of the torque density and used fewer magnets than the initial model. Based on the methodology and findings of this study, the following future research avenues are suggested:

• Consider the non-linear characteristics of magnetic materials;
• Consider the endcap effects for flux leakage prediction;
• Consider the mechanical loss in the optimization and then verify this quantity experimentally;
• Consider the demagnetization risk in the magnets.