# Modeling and Design Optimization of A Shaft-Coupled Motor and Magnetic Gear

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Electromagnetic Modeling

#### 2.1. Harmonic Modeling Method

- The electromagnetic problem can be described in a 2D polar coordinate system ($r,\phi $)
- For a given region, the material has linear and homogenous magnetic properties in the r-direction.
- The ferromagnetic material is infinitely permeable. Consequently, no analytical expression of the magnetic flux density can be obtained within the ferromagnetic material.

#### 2.2. Harmonic Modeling of an Electrical Motor

- Neumann boundary conditionThis specifies the tangential magnetic field strength to be ${H}_{\phi}=0$ [8] at the boundary between two regions of which one has $\mu =\infty $, giving
- Between region I and rotor ferromagnetic core ($r={r}_{1}$)$${B}_{\phi ,I}-{\mu}_{0}{M}_{\phi ,I}=0$$
- Between region IV${}_{q}$ and stator ferromagnetic core ($r={r}_{5}$), for $q=1,...,Q$$${B}_{\phi ,I{V}_{q}}=0$$

- Continuous boundary conditionBy applying Maxwell equations at the interface between different regions [13], the boundary conditions ${B}_{r,k}={B}_{r,k+1}$ and ${H}_{\phi ,k}={H}_{\phi ,k+1}$ are obtained, where k and $k+1$ indicate adjacent regions having a finite permeability ($\mu \ne \infty $). Considering regions with the same tangential width, the following continuous boundary conditions are identified:
- Between regions I and II ($r={r}_{2}$)$${B}_{r,I}={B}_{r,II}$$$${B}_{\phi ,I}-{\mu}_{0}{M}_{\phi ,I}={\mu}_{r,in}{B}_{\phi ,II}$$
- Between regions III${}_{q}$ and IV${}_{q}$ and stator ferromagnetic core ($r={r}_{4}$), for $q=1,...,Q$$${B}_{r,II{I}_{q}}={B}_{r,I{V}_{q}}$$$${B}_{\phi ,II{I}_{q}}={B}_{\phi ,I{V}_{q}}$$

- Combination of Neumann and continuous boundary conditionsEach slot air region III${}_{1,\phantom{\rule{0.277778em}{0ex}}\dots ,\phantom{\rule{0.277778em}{0ex}}Q}$ has different tangential widths with respect to the radially adjacent airgap region II. This gives rise to a combination of both Neumann and continuous boundary conditions which hold at certain intervals between regions II and III${}_{q}$, for $q=1,\phantom{\rule{0.277778em}{0ex}}\dots ,\phantom{\rule{0.277778em}{0ex}}Q$:$${B}_{r,II{I}_{q}}={B}_{r,II},\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}0\le {\phi}_{jq}\le {\alpha}_{s}$$$$\begin{array}{cc}\hfill {H}_{\phi ,II}& =\sum _{q=1}^{Q}{H}_{{\phi}_{II{I}_{q}}},\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}0\le {\phi}_{jq}\le {\alpha}_{s},\hfill \\ & =0,\phantom{\rule{1.em}{0ex}}\mathrm{elsewhere}\hfill \end{array}$$The slots introduce tangential Neumann boundary conditions on region III${}_{q}$, resulting in ${H}_{r}=0$ at the tangential boundaries ${\phi}_{jq}=0$ and ${\phi}_{jq}={\alpha}_{s}$. Consequently, the cosine component of ${B}_{r}$ in in region III${}_{q}$ vanishes, resulting in ${B}_{rc}=0$ and ${B}_{\phi s}=0$. Therefore, the boundary condition Equation (20) can be expressed as$$\sum _{m=1}^{M}{B}_{rs,II{I}_{q}}sin\left(\frac{m\pi}{({\alpha}_{s}/2)}{\phi}_{jq}\right)=\sum _{n=1}^{N}{B}_{rs,II}sin\left(\frac{n\pi}{{\tau}_{k}}{\phi}_{k}\right)$$$${B}_{rs,II{I}_{q}}=\sum _{n=1}^{N}({B}_{rs,II}{\u03f5}_{s}+{B}_{rc,II}{\u03f5}_{c})$$The boundary condition Equation (21) can be expressed as$$\begin{array}{c}\hfill \sum _{n=1}^{N}\left[\left({B}_{\phi {s}_{II}}-{\mu}_{0}{M}_{\phi {s}_{II}}\right)sin\left(\frac{n\pi}{{\tau}_{k}}{\phi}_{II}\right)+\left({B}_{\phi {c}_{II}}-{\mu}_{0}{M}_{\phi {c}_{II}}\right)cos\left(\frac{n\pi}{{\tau}_{k}}{\phi}_{II}\right)\right]=\\ \hfill \sum _{q=1}^{Q}\left[\sum _{m=1}^{M}\left[\left({B}_{\phi c,II{I}_{q}}-{\mu}_{0}{M}_{\phi c,II{I}_{q}}\right)cos\left(\frac{m\pi}{({\alpha}_{s}/2)}{\phi}_{jq}\right)+{B}_{\phi 0,II{I}_{q}}\right]\right]\end{array}$$$${B}_{\phi {s}_{II}}-{\mu}_{0}{M}_{\phi {s}_{II}}=\sum _{q=1}^{Q}\left[\sum _{m=1}^{M}\left[\left({B}_{\phi c,II{I}_{q}}-{\mu}_{0}{M}_{\phi c,II{I}_{q}}\right){\kappa}_{c}+{B}_{\phi 0,II{I}_{q}}{\kappa}_{0}\right]\right]$$$${B}_{\phi {c}_{II}}-{\mu}_{0}{M}_{\phi {c}_{II}}=\sum _{q=1}^{Q}\left[\sum _{m=1}^{M}\left[\left({B}_{\phi c,II{I}_{q}}-{\mu}_{0}{M}_{\phi c,II{I}_{q}}\right){\zeta}_{c}+{B}_{\phi 0,II{I}_{q}}{\zeta}_{0}\right]\right]$$

#### 2.2.1. Model Verification

#### 2.3. Harmonic Modeling of a Magnetic Gear

- Neumann boundary condition at $r={r}_{1}$ and $r={r}_{6}$
- Continuous boundary condition at $r={r}_{2}$ and $r={r}_{5}$
- Combination of Neumann and continuous boundary conditions at $r={r}_{3}$ (between regions II and III${}_{1,\phantom{\rule{0.277778em}{0ex}}\dots ,\phantom{\rule{0.277778em}{0ex}}Q}$) and $r={r}_{4}$ (between regions IV and III${}_{1,\phantom{\rule{0.277778em}{0ex}}\dots ,\phantom{\rule{0.277778em}{0ex}}Q}$)
- Conservation of the magnetic flux around the pole piecesThis boundary condition concerns Gauss’ law for magnetic field given by$${\oint}_{S}\overrightarrow{B}ds=0$$By applying the the above to the pole-piece depicted in Figure 8, the following is obtained$$\begin{array}{cc}& {\int}_{{\phi}_{q}}^{{\phi}_{q+1}}{B}_{r,IV}{|}_{r={r}_{4}}-{\int}_{{\phi}_{q}}^{{\phi}_{q+1}}{B}_{r,II}{|}_{r={r}_{3}}+\hfill \\ & {\int}_{{r}_{3}}^{{r}_{4}}{B}_{\phi ,II{I}_{q+1}}{|}_{\phi ={\phi}_{q+1}}-{\int}_{{r}_{3}}^{{r}_{4}}{B}_{\phi ,II{I}_{q}}{|}_{\phi ={\phi}_{q}}=0\hfill \end{array}$$$$\underset{{h}_{c}\to 0}{lim}{\oint}_{C}\overrightarrow{H}dl={\int}_{S}\overrightarrow{J}ds$$$$\sum _{q=1}^{Q}{\alpha}_{pp}{H}_{\phi ,II{I}_{q}}=2\pi {H}_{\phi ,IV}$$$$\sum _{q=1}^{Q}\frac{{\alpha}_{pp}}{{\mu}_{0}}{B}_{\phi 0,II{I}_{q}}=0$$

#### 2.3.1. Model Verification

## 3. Definition of Optimization Problem Statements

#### 3.1. Optimization Problem Statement for the Electrical Motor

- Objective functionThe considered application requires that the motor and magnetic gear are compact and lightweight. An objective function that handles these requirements is the inverse of the mass torque density$${f}_{mot}({\overrightarrow{X}}_{mot})=\frac{{m}_{mot}}{{\widehat{T}}_{mot}}$$The motor torque ${T}_{mot}$ can be calculated using Maxwell stress tensor as follows:$${T}_{mot}=\frac{L{r}_{in}^{2}}{{\mu}_{0}}{\int}_{0}^{2\pi}{B}_{r,II}{B}_{\phi ,II}d\phi $$
- Inequality constraint functions
- –
- Winding temperatureBased on Equation (36), the following constraint on the peak torque is defined$${g}_{1,mot}={\Theta}_{w,10s}\le 100{\phantom{\rule{0.277778em}{0ex}}}^{\circ}C\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{at}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\widehat{P}}_{w}$$$$\left[\begin{array}{cc}\frac{1}{{R}_{tw}+{R}_{y,1}}& -\frac{1}{{R}_{tw}+{R}_{y,1}}\\ -\frac{1}{{R}_{tw}+{R}_{y,1}}& \frac{1}{{R}_{tw}+{R}_{y,1}}+\frac{1}{{R}_{y,2}+{R}_{conv}}\end{array}\right]\left[\begin{array}{c}{\Theta}_{w}\\ {\Theta}_{s}\end{array}\right]+\frac{d}{dt}\left[\begin{array}{cc}{C}_{w}& 0\\ 0& {C}_{s}\end{array}\right]\left[\begin{array}{c}{\Theta}_{w}\\ {\Theta}_{s}\end{array}\right]=\left[\begin{array}{c}{P}_{w}\\ \frac{{\Theta}_{amb}}{{R}_{y,2}+{R}_{h}}\end{array}\right]$$$${P}_{w}=\frac{Q{(J/\sqrt{2})}^{2}{\rho}_{cu}{l}_{coil}{S}_{coil}}{{k}_{f}}$$
- –
- Magnetic flux density in the ferromagnetic coresThe used ferromagnetic core material has a typical saturation point at B = 1.5 T in its $B-H$ characteristic. Thus, to maintain a linear current-torque relation in the motor, the following constraint functions are defined$${g}_{2,mot}({\overrightarrow{X}}_{mot})={B}_{t}\le 1.5\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$$$${g}_{3,mot}({\overrightarrow{X}}_{mot})={B}_{s}\le 1.5\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$$$${B}_{ti}=\frac{{\varphi}_{ti}}{L{w}_{t}}$$$${B}_{sj}=\frac{{\varphi}_{si}}{L({r}_{6}-{r}_{5})}$$$${\varphi}_{ti}=L{r}_{3}{\int}_{{\phi}_{i,1}}^{{\phi}_{i,2}}{B}_{II,r}d\phi $$$${\varphi}_{sj}=-\frac{1}{Q}\sum _{j=1}^{Q-1}{\varphi}_{t\mathrm{mod}(j+i-1,Q)}$$
- –
- Torque rippleA smooth torque characteristic is required in the considered application. For that reason, the ripple in the motor torque shown Figure 13 is constrained by the following function$${g}_{4,mot}({\overrightarrow{X}}_{mot})=\frac{max({\widehat{T}}_{mot})-min({\widehat{T}}_{mot})}{{\widehat{T}}_{mot,avg}}\le 1\phantom{\rule{0.277778em}{0ex}}\%$$

- Equality constraint functionsA series of optimization tasks will be performed on the motor. For a given optimization task, fixed values of motor outer dimensions are assigned. Therefore, the following equality constraint on outer diameter ${D}_{mot}$ is introduced$${h}_{1,mot}({\overrightarrow{X}}_{mot})={D}_{mot}$$
- Design variables and bound constraintsThe design variable vector ${\overrightarrow{X}}_{mot}$ consists of motor geometric parameters (see Figure 4) and current density$${\overrightarrow{X}}_{mot}=[{r}_{0},({r}_{1}-{r}_{0}),({r}_{2}-{r}_{1}),({r}_{3}-{r}_{2}),({r}_{4}-{r}_{3}),({r}_{5}-{r}_{4}),({r}_{6}-{r}_{5}),{\alpha}_{s},{\tau}_{m},{w}_{t},J]$$$${\overrightarrow{X}}_{mot}^{l}=\left[1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},0.2\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},0.2\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},{3}^{\circ},0.5,1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},\frac{J}{10}\right]$$$${\overrightarrow{X}}_{mot}^{u}=\left[\frac{{r}_{6}}{2},\frac{{r}_{6}}{2},\frac{{r}_{6}}{2},\frac{{r}_{6}}{2},\frac{{r}_{6}}{2},\frac{{r}_{6}}{2},{15}^{\circ},\frac{{r}_{6}}{2},1,1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},10J\right]$$

#### 3.2. Optimization Problem Statement for the Magnetic Gear

- Objective functionSimilar to the electrical motor optimization, the defined objective function of the magnetic gear is the inverse of mass torque density$${f}_{mg}({\overrightarrow{X}}_{mg})=\frac{{m}_{mg}}{{\widehat{T}}_{mg,out}}$$$$\begin{array}{cc}\hfill {T}_{mg,out}& ={T}_{mg,stat}-{T}_{mg,in}\hfill \\ & =\frac{L{\left(\frac{{r}_{4}+{r}_{5}}{2}\right)}^{2}}{{\mu}_{0}}{\int}_{0}^{2\pi}{B}_{r,IV}{B}_{\phi ,IV}d\phi -\frac{L{\left(\frac{{r}_{2}+{r}_{3}}{2}\right)}^{2}}{{\mu}_{0}}{\int}_{0}^{2\pi}{B}_{r,II}{B}_{\phi ,II}d\phi \hfill \end{array}$$
- Inequality constraint functions
- –
- Magnetic flux density in the ferromagnetic coresConstraints on the magnetic flux density in the pole-pieces and stator core of the magnetic gear are introduced to avoid saturation in the ferromagnetic cores, which leads to the inaccuracy of the analytical model with respect to the FEM model that accounts for nonlinear $B-H$ curve of the ferromagnetic steel 1010. The constraint values are selected such that the resulting torque is maximized while the analytical model accuracy is not significantly sacrificed. Figure 15 shows the variations of torque and discrepancy between analytical and FEM models for different values of the constraints ${B}_{sat,PP}$ and ${B}_{sat,SC}$, belonging to the pole-pieces and stator core, respectively. The constraints ${B}_{sat,PP}=3$ T and ${B}_{sat,PP}=3$ T as calculated by the analytical model are selected based on the previous consideration on torque and model accuracy; thus, the following constraint functions are defined:$${g}_{1,mg}({\overrightarrow{X}}_{mg})={B}_{SC}\le 3\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$$$${g}_{2,mg}({\overrightarrow{X}}_{mg})={B}_{PP}\le 3\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$$$${B}_{SC}=\frac{{\varphi}_{SC}}{L({r}_{7}-{r}_{6})}=\frac{L{r}_{6}{\int}_{0}^{{\alpha}_{SC}}{B}_{r,V}d\phi}{L({r}_{7}-{r}_{6})}$$$${B}_{PP}=\sqrt{{B}_{r,PP}^{2}+{B}_{\phi ,PP}^{2}}=\sqrt{{\left(\frac{1}{{r}_{PP}}\frac{\partial {A}_{z,PP}}{\partial \phi}\right)}^{2}+{\left(-\frac{\partial {A}_{z,PP}}{\partial r}\right)}^{2}}$$$${A}_{z,PP}={A}_{z,II}\frac{{r}_{4}-{r}_{PP}}{{r}_{4}-{r}_{3}}+{A}_{z,IV}\frac{{r}_{PP}-{r}_{3}}{{r}_{4}-{r}_{3}}$$

- Equality constraint functionsFollowing Equation (48), the following constraint on the magnetic gear outer diameter ${D}_{mg}$ is introduced$${h}_{1,mg}({\overrightarrow{X}}_{mg})={D}_{mg}$$Additionally, the magnetic gear axial length ${D}_{mg}$ is fixed as a design parameter.
- Design variables and bound constraintsThe design variable vector ${\overrightarrow{X}}_{mg}$ consists of the magnetic gear geometric parameters (see Figure 7)$${\overrightarrow{X}}_{mg}=[{r}_{0},({r}_{1}-{r}_{0}),({r}_{2}-{r}_{1}),({r}_{3}-{r}_{2}),({r}_{4}-{r}_{3}),({r}_{5}-{r}_{4}),({r}_{6}-{r}_{5}),({r}_{7}-{r}_{6}),{\tau}_{Q},{\tau}_{m,out}]$$$${\overrightarrow{X}}_{mg}^{l}=[1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},0.2\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},0.2\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},0.3,0.3]$$$${\overrightarrow{X}}_{mg}^{u}=\left[\frac{{r}_{7}}{2},\frac{{r}_{7}}{2},\frac{{r}_{7}}{2},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},\frac{{r}_{7}}{2},1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm},\frac{{r}_{7}}{2},\frac{{r}_{7}}{2},0.7,1\right]$$

## 4. Optimization of the Shaft-Coupled Electrical Motor and Magnetic Gear

#### 4.1. Modeling

#### 4.2. Design Requirements

- The diameter of motor is equal to that of magnetic gear.
- The maximum axial length of the actuator is twice its diameter.

#### 4.3. Response Surface Methodology

- 20 mm $\le {D}_{mot}\le $ 40 mm, 5 mm $\le {L}_{mot}\le $ 15 mm
- 20 mm $\le {D}_{mg}\le $ 40 mm, 10 mm $\le {L}_{mg}\le $ 20 mm, 3.33 $\le {N}_{mg}\le $ 9.67.

#### 4.4. Definition of Optimization Problem Statement

#### 4.5. Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Integration possibilities of electrical motor and magnetic gear: (

**a**) shaft-coupled motor and gear; (

**b**) “pseudo” direct-drive motor.

**Figure 11.**Motor winding and stator temperature dynamic responses for changing values of copper loss, at ambient temperature of 20 ${}^{\circ}$C.

**Figure 14.**Torque exerted on the the magnetic gear rotors and stator as a function of inner rotor position ${\theta}_{mg,in}$, while the outer rotor position is fixed.

**Figure 15.**Optimized magnetic gear outer rotor peak torque ${\widehat{T}}_{mg,out}$ and its estimation error, as functions of magnetic flux density saturation constraints in the pole-pieces and stator core, ${B}_{sat,PP}$ and ${B}_{sat,SC}$, respectively.

**Figure 16.**The circled zero crossing of ${A}_{z,V}$ indicates the angle ${\alpha}_{SC}$, over which flux passes through the stator core.

**Figure 17.**Estimated magnetic flux densities in the stator core ${B}_{SC}$ and pole-pieces ${B}_{PP}$ as a function of inner rotor position ${\theta}_{mg,in}$.

**Figure 18.**Optimized volume torque density characteristic of the magnetic gear as a function of its transmission ratio.

**Figure 21.**Optimized peak motor torque ${\widehat{T}}_{mot}$ and magnetic gear outer rotor torque ${T}_{mg,out}$ characteristics.

**Table 1.**Modeled regions in the motor, based on Figure 4.

Region | Description | Parameters |
---|---|---|

I | PM array | - Number of PM pole pairs, p |

- Pole-arc to pole-pitch ratio, ${\tau}_{m}=\frac{{\alpha}_{1}}{{\alpha}_{2}}$ | ||

- Remanence, ${B}_{r}$ | ||

- Relative permeability, ${\mu}_{r}$ | ||

II | Airgap | N/A |

III${}_{1,\phantom{\rule{0.277778em}{0ex}}\dots ,\phantom{\rule{0.277778em}{0ex}}Q}$ | Slot air | - Number of slots, Q |

- Slot opening, ${\alpha}_{s}$ | ||

IV${}_{1,\phantom{\rule{0.277778em}{0ex}}\dots ,\phantom{\rule{0.277778em}{0ex}}Q}$ | Slot winding | - Number of slots, Q |

- Slot opening, ${\alpha}_{s}$ | ||

- Current density, J |

Parameter | Value |
---|---|

No. of PM pole pairs, p | 7 |

No. of slots, Q | 12 |

Inner shaft radius, ${r}_{0}$ | 2.7 mm |

Inner ferromagnetic core outer radius, ${r}_{1}$ | 3.7 mm |

Inner PM outer radius, ${r}_{2}$ | 5.7 mm |

Inner airgap outer radius, ${r}_{3}$ | 6 mm |

Slot air outer radius, ${r}_{4}$ | 6.5 mm |

Slot winding outer radius, ${r}_{5}$ | 10.5 mm |

Outer stator radius, ${r}_{6}$ | 12.5 mm |

Axial length, L | 18 mm |

Slot opening, ${\alpha}_{s}$ | 5${}^{\circ}$ |

Tooth width, ${w}_{t}$ | 2 mm |

PM pole-arc to pole-pitch ratio, ${\tau}_{m}$ | 1 |

PM remanence, ${B}_{r}$ | 1.39 |

PM relative permeability, ${\mu}_{r}$ | 1.05 |

Ferromagnetic core material (for FEM) | M330-35A |

Current density, J | 5 A/mm${}^{2}$ |

**Table 3.**Simulation duration of the analytical and FEM (Finite Element Method) models, as executed in a PC with Intel Core i5-2500 (3.3 GHz) processor, 16 GB RAM and 64-bit Windows 7 OS.

Duration | ||
---|---|---|

Analytical (Linear) | FEM (Linear) | FEM (Nonlinear) |

0.05 s | 1 s | 6 s |

Region | Description | Parameters |
---|---|---|

I | Inner PM array | - Number of inner PM pole pairs, ${p}_{in}$ |

- Pole-arc to pole-pitch ratio, ${\tau}_{m,in}=\frac{{\alpha}_{1}}{{\alpha}_{2}}$ | ||

- Remanence, ${B}_{r}$ | ||

- Relative permeability, ${\mu}_{r}$ | ||

II | Inner airgap | N/A |

III${}_{1,\phantom{\rule{0.277778em}{0ex}}\dots ,\phantom{\rule{0.277778em}{0ex}}Q}$ | Air between pole-pieces | - Number of pole-pieces, Q |

- Tangential width, ${\alpha}_{pp}$ | ||

- Pole-piece arc-to-pitch ratio, ${\tau}_{Q}=1-\frac{{\alpha}_{pp}}{2\pi /Q}$ | ||

IV | Outer airgap | N/A |

V | Outer PM array | - Number of outer PM pole pairs, ${p}_{out}$ |

- Pole-arc to pole-pitch ratio, ${\tau}_{m,out}=\frac{{\alpha}_{3}}{{\alpha}_{4}}$ | ||

- Remanence, ${B}_{r}$ | ||

- Relative permeability, ${\mu}_{r}$ |

Parameter | Value |
---|---|

No. of inner PM pole pairs, ${p}_{in}$ | 2 |

No. of outer PM pole pairs, ${p}_{out}$ | 5 |

No. of pole-pieces, $Q={p}_{in}+{p}_{out}$ | 7 |

Transmission ratio, ${N}_{mg}=\frac{Q}{{P}_{in}}$ | 3.5 |

Inner shaft radius, ${r}_{0}$ | 2.5 mm |

Inner ferromagnetic core outer radius, ${r}_{1}$ | 4.5 mm |

Inner PM outer radius, ${r}_{2}$ | 5.5 mm |

Inner airgap outer radius, ${r}_{3}$ | 6 mm |

Pole-piece outer radius, ${r}_{4}$ | 8.5 mm |

Outer airgap outer radius, ${r}_{5}$ | 9 mm |

Outer PM outer radius, ${r}_{6}$ | 10 mm |

Stator outer radius, ${r}_{7}$ | 12.5 mm |

Inner PM pole-arc to pole-pitch ratio, ${\tau}_{m,in}$ | 1 |

Outer PM pole-arc to pole-pitch ratio, ${\tau}_{m,out}$ | 0.9 |

Pole-piece arc to pitch ratio, ${\tau}_{Q}$ | 0.5 |

PM remanence, ${B}_{r}$ | 1.39 |

PM relative permeability, ${\mu}_{r}$ | 1.05 |

Ferromagnetic core material (for FEM) | Steel 1010 |

**Table 6.**Comparison between outer dimensions and volume of the shaft-coupled motor and magnetic gear, and an electrical motor.

Parameter | Shaft-Coupled Motor and Magnetic Gear | Electrical Motor Only |
---|---|---|

Diameter | 24 mm | 30 mm |

Axial length (incl. housing) | 48 mm | 45 mm |

Volume | 2.2$\times {10}^{4}$ mm${}^{3}$ | 3.2$\times {10}^{4}$ mm${}^{3}$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zanis, R.; Jansen, J.W.; Lomonova, E.A. Modeling and Design Optimization of A Shaft-Coupled Motor and Magnetic Gear. *Actuators* **2016**, *5*, 10.
https://doi.org/10.3390/act5010010

**AMA Style**

Zanis R, Jansen JW, Lomonova EA. Modeling and Design Optimization of A Shaft-Coupled Motor and Magnetic Gear. *Actuators*. 2016; 5(1):10.
https://doi.org/10.3390/act5010010

**Chicago/Turabian Style**

Zanis, R., J.W. Jansen, and E.A. Lomonova. 2016. "Modeling and Design Optimization of A Shaft-Coupled Motor and Magnetic Gear" *Actuators* 5, no. 1: 10.
https://doi.org/10.3390/act5010010