# Comparative Study of the Dual Layer Magnet Array in a Moving-Coil Tubular Linear PM Motor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Topologies of Magnetic Field Distribution

- (1)
- The axial length of the motor is infinite along the z axis and the motor’s structure is axially symmetric and periodic in the z direction. The edge effects associated with the finite length of the motor will be considered by studying a whole model which includes multiple such length-limited motors with enough long distance between any adjacent two motors for convenient Fourier expansion [29].
- (2)
- The permeability of the iron is infinite, which means saturation is ignored.
- (3)
- The armature is slotless. However, slotting effects, if present, can be taken into account by introducing a Carter coefficient [30].

**M**is the magnetization intensity of magnet. For a permanent magnet that has a linear demagnetization characteristic, ${\mu}_{r}$ is constant and the remanent magnetization

**M**is related to the remanence

**${\mathbf{B}}_{rem}$**by

**B**satisfies

**A**is introduced in terms of the Coulomb gauge as

**A**can be decomposed as $\mathbf{A}={A}_{r}{\mathbf{e}}_{\mathbf{r}}+{A}_{\theta}{\mathbf{e}}_{\theta}+{A}_{z}{\mathbf{e}}_{\mathbf{z}}$ and

**M**is given by

**A**only has the component of ${A}_{\theta}$ that is independent of $\theta $. The following Equation (7) includes governing equations of Regions 1, 2 and 3, respectively.

#### 2.1. Dual Layer Halbach Magnetization Topology

**M**, is given by

#### 2.2. Dual Layer Radial Magnetization Topology

**M**is zero. As a result, the boundary conditions are the same as Equation (12). In other words, the governing field equations and the boundary conditions are completely the same as the ones of dual Halbach magnetic array. The flux density distributions are thus given by Equations (13)–(15) similarly.

#### 2.3. Dual Layer Axial Magnetization Topology

#### 2.4. Axial-Halbach Magnetization Topology

#### 2.5. Radial-Halbach Magnetization

**M**is given by

#### 2.6. Axial-Radial Magnetization Topology

## 3. Numerical Simulation and Experiments

#### 3.1. Magnetic Field Variation of the Topologies

#### 3.2. Prototype and Experimental Apparatus

#### 3.3. Validation of Analytical Magnetic Field Model

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FEM | Finite element methom |

PM | Permanent magnet |

PC | Personal computer |

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**Figure 1.**Typical single layer magnet array topologies of tubular linear PM motor: (

**a**) axial magnetization; (

**b**) radial magnetization; (

**c**) Halbach magnetization; and (

**d**) quasi-Halbach magnetization.

**Figure 2.**Topologies of double-layer magnetic array: (

**a**) dual layer axial magnetization; (

**b**) dual layer radial magnetization; (

**c**) dual layer quasi-Halbach magnetization; (

**d**) dual layer with quasi-Halbach array and radial magnet; (

**e**) dual layer with quasi-Halbach magnetization and axial magnet; and (

**f**) dual layer with radial magnetization and axial magnetization.

**Figure 4.**Field regions of dual layer quasi-Halbach magnetization array topology: (

**a**) solving regions, (

**b**,

**c**) magnetization distributions.

**Figure 5.**Filed regions of dual layer radially magnetized array topology: (

**a**) field regions and (

**b**) magnetization distributions.

**Figure 6.**Filed regions of dual layer axially magnetized array topology: (

**a**) field regions and (

**b**) magnetization distributions.

**Figure 7.**Field regions of dual layer axial-Halbach magnetization topologies: (

**a**,

**c**) solving regions of two topologies; (

**b**,

**d**) magnetization distributions.

**Figure 8.**Field regions of dual layer radial-Halbach magnetization topologies: (

**a**,

**b**) solving regions of two topologies, (

**c**,

**d**) magnetization distributions.

**Figure 9.**Field regions of dual layer axial-radial magnetization topologies: (

**a**,

**c**) solving regions of two topologies; (

**b**,

**d**) magnetization distributions.

**Figure 10.**Topology of fresh model which is established for analysis of edgy effects of dual layer quasi-Halbach magnetization array.

**Figure 12.**Linear machines with dual Halbach array. (

**a**) Research prototype; (

**b**) Experimental testbed.

Motor length L | 250 mm |

Maximum radius ${R}_{o}$ | 50 mm |

Width of radial PM ${\tau}_{r}$ | 5 mm |

Pole-pitch ${\tau}_{p}$ | 25 mm |

Number of poles n | 9 |

Air gap length g | 1 mm |

Outer rad of external PM ${R}_{s}$ | 45 mm |

Inner rad of external PM ${R}_{b}$ | 32 mm |

Outer rad of internal PM ${R}_{a}$ | 23 mm |

Inner rad of internal PM ${R}_{r}$ | 13 mm |

Number of winding turns | 60 |

No. | FEM (T) | Analytical Method (T) | Experimental Method (T) |
---|---|---|---|

1 | −0.7324 | −0.7982 | −0.7183 |

2 | −0.7882 | −0.7830 | −0.7725 |

3 | −0.7910 | −0.8014 | −0.7882 |

4 | −0.6902 | −0.7025 | −0.6216 |

5 | −0.1413 | −0.1408 | −0.1425 |

6 | 0.3922 | 0.3919 | 0.3936 |

7 | 0.8381 | 0.8205 | 0.8194 |

8 | 0.7914 | 0.7944 | 0.7923 |

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**MDPI and ACS Style**

Yan, L.; Zhang, L.; Peng, L.; Jiao, Z.
Comparative Study of the Dual Layer Magnet Array in a Moving-Coil Tubular Linear PM Motor. *Sensors* **2018**, *18*, 1854.
https://doi.org/10.3390/s18061854

**AMA Style**

Yan L, Zhang L, Peng L, Jiao Z.
Comparative Study of the Dual Layer Magnet Array in a Moving-Coil Tubular Linear PM Motor. *Sensors*. 2018; 18(6):1854.
https://doi.org/10.3390/s18061854

**Chicago/Turabian Style**

Yan, Liang, Lu Zhang, Lei Peng, and Zongxia Jiao.
2018. "Comparative Study of the Dual Layer Magnet Array in a Moving-Coil Tubular Linear PM Motor" *Sensors* 18, no. 6: 1854.
https://doi.org/10.3390/s18061854