A single-phase tubular permanent-magnet linear machine (PMLM) with hybrid Halbach/axially-magnetized PM arrays is proposed for free-piston Stirling power generation system. Machine topology and operating principle are elaborately illustrated. With the sinusoidal speed characteristic of the free-piston Stirling engine considered, the proposed machine is designed and calculated by finite-element analysis (FEA). The main structural parameters, such as outer radius of the mover, radial length of both the axially-magnetized PMs and ferromagnetic poles, axial length of both the middle and end radially-magnetized PMs, etc., are optimized to improve both the force capability and power density. Compared with the conventional PMLMs, the proposed machine features high mass and volume power density, and has the advantages of simple control and low converter cost. The proposed machine topology is applicable to tubular PMLMs with any phases.

## I. INTRODUCTION

Free-piston Stirling power generation system, which integrates a free-piston Stirling engine with a tubular linear machine, is increasingly investigated for applications where require quiet operation, high reliability and long service life.^{1,2} Owing to the advantages of high power density and high efficiency, permanent-magnet linear machines (PMLMs) are inherently suitable for the Stirling generation system.^{3,4} There are three kinds of mover topologies for conventional tubular PMLMs, i.e., movers with radially-, axially-, and Halbach magnetized permanent magnets (PMs).^{5,6} Structure of the radially-magnetized PM mover is relatively simple, but it needs thick mover core, which leads to large mover mass. For the axially-magnetized PMLM, the air-gap flux density is increased due to flux concentration effect of the ferromagnetic poles, but there is leakage flux in the inner side of the mover, and the leakage flux sharply increases when the air-gap length becomes larger.^{7} The Halbach PM array is beneficial for obtaining high air-gap flux density and sinusoidal magnetic field distribution. While the homopolar radially-magnetized PM ring is difficult to manufacture, so the homopolar PM ring is generally assembled by several cambered PM segments in the manufacturing process, and these PM segments are banded by fabric bandages, which increases the equivalent air-gap length and compromises the machine performance.^{8,9}

Herein, we propose a single-phase tubular PMLM with hybrid Halbach/axially-magnetized PMs. The proposed machine has the advantages of both the axially-magnetized and Halbach PMLMs, such as high air-gap flux density. Meanwhile, it eliminates the leakage flux problem inside the axially-magnetized PMLM and solves the fixation problem of the cambered radially-magnetized PM segments in the Halbach PMLM. This paper focuses on the design and optimization of the proposed machine.

## II. MACHINE STRUCTURE AND OPERATING PRINCIPLE

The proposed machine consists of two cylindrical components, *i.e.*, stator and mover, as shown in Figure 1(a)–(b). The stator has a slotted stator core, and an annular stator winding is inserted in the stator slot. The mover is composed of radially- and axially-magnetized PM rings, ferromagnetic poles, and titanium tube. Differing from the axially-magnetized PM ring, the radially-magnetized PM ring is generally assembled by several cambered PM segments. The radially-magnetized PM rings are inset in the bore of the ferromagnetic poles, and the axially-magnetized PMs are sandwiched between the ferromagnetic poles. These ferromagnetic poles not only serve as magnetic paths of both the radially- and axially-magnetized PMs, but also confine the radial movement of the radially-magnetized PM segments. Hence, no fabric bandages are needed for the mover fixation of the proposed machine, and the equivalent air-gap length of the proposed machine can be smaller than that of the conventional Halbach PMLM, which means higher air-gap flux density and higher power density can be obtained for the proposed machine. Since the mover of the proposed machine can be considered as an axially-magnetized PM mover whose bore embeds a Halbach array, the proposed machine is named as hybrid Halbach/axially-magnetized PMLM. Considering the weight and robustness of the mover, the titanium tube is selected to support the whole mover.

When the mover is at two different limiting positions, the vector plots of no-load flux distribution are shown in Figure 1(c)–(d). As the mover moves from the top limiting position to the bottom one, the flux linkage that links the stator winding varies, meanwhile its flowing direction changes from one direction to the opposite one. When the mover oscillates with piston of the Stirling engine, alternate voltage is induced in the stator winding.

## III. ELECTROMAGNETIC PERFORMANCE OPTIMIZATION

The proposed machine is designed and calculated by finite-element analysis (FEA). With the speed characteristic of free-piston Stirling engine considered, a sinusoidal speed is applied to the mover in the FEA model. Force capability and power density are key performance parameters for the PMLM used in the Stirling generation system, and they are mainly influenced by some main structural parameters, such as the outer radius of the mover *R*_{m}, the air-gap length *δ*, the radial length of both the axially-magnetized PMs and ferromagnetic poles (*h*_{M} and *h*_{Fe}), the axial length of both the middle and end radially-magnetized PMs (*b*_{Mc} and *b*_{Me}), the pole pitch of the mover and stator ($\tau pm$ and$\tau ps$), the stator tooth width *w*_{ts}, and the stator yoke width *w*_{ys}. Due to system requirements and manufacturing consideration, the out diameter and air-gap length are selected to be 100mm and 0.8mm, respectively. To reduce the computation work, some structural parameters, which play more important role on the machine performance, are firstly optimized.

The average force and power density with respect to the outer radius of the mover *R*_{m} are calculated by FEA, as shown in Figure 2(a). With the increase of *R*_{m}, both the force and power density first increase and then decrease. The force and power density achieve maximum values when *R*_{m} equals 23.2mm and 24.2mm, respectively. To obtain high power density, *R*_{m} is chosen to be 24.2mm, and this optimum air-gap diameter indicates a good balance between electric and magnetic loadings.

With the optimum *R*_{m} of 24.2mm, the average force and power density with respect to the radial length of the axially-magnetized PMs *h*_{M} are calculated, as shown in Figure 2(b). When *h*_{M} increases, the magneto motive force (MMF) and magnetic flux that provided by the mover increase, so both the force and power density increase with the increase of *h*_{M}. As *h*_{M} is above 12mm, the power density almost keeps unchanged with the increase of *h*_{M}, hence, *h*_{M} is selected to be 12mm.

By choosing different values of *b*_{mc}/*τ*_{pm} and maintaining $\tau pm$ unchanged, the average force and power density are calculated, as shown in Figure 2(c). With the increase of *b*_{mc}/*τ*_{pm}, the axial lengths of both the middle ferromagnetic pole and radially-magnetized PM increase. Then the leakage flux that circulates between the axially-magnetized PMs, stator tooth tip, and both the middle and end ferromagnetic poles increase, and the middle ferromagnetic pole is more subject to saturation, further leading to the reduction of force capability. When *b*_{mc}/*τ*_{pm} equals 0.64, the power density achieves maximum value. Hence, *b*_{mc}/*τ*_{pm} is chosen to be 0.64.

To reduce the manufacturing workload, radial lengths of both the middle and end ferromagnetic poles are suggested to be the same. The average force and power density with respect to the radial length of ferromagnetic poles *h*_{Fe} are calculated, as shown in Figure 2(d). Since magnetic flux of both the radially- and axially-magnetized PMs flows through the middle ferromagnetic pole, the middle ferromagnetic pole is more subject to saturation when *h*_{Fe} is small, hence both the force and power density are low. With the increase of *h*_{Fe}, the saturation in the middle ferromagnetic weakens, both the force and power density increase. As *h*_{Fe} is relatively large, the leakage flux which is due to the axially-magnetized PMs sharply increases in the inner side of the mover, hence both the force and power density decrease. When *h*_{Fe} equals 6mm, both the force and power density achieve maximum value, hence, *h*_{Fe} is chosen to be 6mm. Besides, when *h*_{Fe} is 12mm, the machine becomes an axially-magnetized PMLM, and both the force and power density are lower than that of the proposed machine.

By choosing different pole pitch of the mover $\tau pm$, both the force and power density are calculated, as shown in Figure 3(a). With the increase of $\tau pm$, both the force and power density increase. When $\tau pm$ is above 23mm, the power density almost keeps unchanged. Hence, $\tau pm$ is chosen to be 23mm, and this value equals the pole pitch of stator $\tau ps$, which indicates the machine achieves maximum power density when pole pitch of both the stator and mover are equal.

Both the average force and power density with respect to the axial length of the end radially-magnetized PMs *b*_{Me} are calculated, as shown in Figure 3(b). The force increases with the increase of *b*_{Me}, while the power density first increases and then reduces, when *b*_{Me} equals 5mm, the power density achieves maximum value. Hence, *b*_{Me} is chosen to be 5mm.

Figure 3(c) shows the variation of both the average force and power density with respect to the stator tooth width *w*_{ts}. With the increase of *w*_{ts}, both the force and power density first increase and then reduce, when *w*_{ts} equals 4mm and 5mm, the force and power density achieve maximum values, respectively. With both the force and power density considered, *w*_{ts} is chosen to be 4.5mm.

Through choosing different stator yoke width *w*_{ys}, both the average force and power density are calculated, as shown in Figure 3(d). Both the force and power density decrease with the increase of *w*_{ys}, when *w*_{ys} is too small, the flux density in the stator yoke is relatively high, which compromises machine efficiency. Considering the tradeoff between different performance parameters, *w*_{ys} is chosen to be 4mm.

## IV. PERFORMANCE COMPARISON WITH OTHER PMLMS

The proposed machine is compared with two three-phase PMLMs with radially- and axially-magnetized PM movers, respectively. The comparison details are listed in Table I. In comparison with the two PMLMs, the proposed machine features advantages of high mass and volume power density, and this is due to the introduction of the proposed machine topology and the increased use of PMs. Besides, the proposed sing-phase machine features simple control and low converter cost.

Parameters . | Radially magnetized^{10}
. | Axially magnetized^{10}
. | Proposed machine . |
---|---|---|---|

Phase number | Three | Three | Single |

Power (kW) | 0.988 | 1.006 | 0.64 |

Current density (A/mm^{2}) | 7.701 | 7.701 | 7.8 |

Average speed (m/s) | 2.8 | 2.8 | 2.8 |

Frequency (Hz) | 100 | 100 | 100 |

Stroke (mm) | 14 | 14 | 14 |

Stator outer diameter (mm) | 100 | 100 | 100 |

Air gap length (mm) | 0.8 | 0.8 | 0.8 |

Axial length of stator (mm) | 63 | 63 | 27.5 |

Axial length of mover (mm) | 81 | 81 | 41.3 |

Stator core mass (kg) | 0.831 | 0.831 | 0.576 |

Copper mass (kg) | 0.842 | 0.842 | 0.393 |

Mover core mass (kg) | 0.491 | 0.385 | 0.055 |

PM mass (kg) | 0.622 | 0.346 | 0.373 |

Mover mass (kg) | 1.113 | 0.731 | 0.428 |

Total mass (kg) | 2.786 | 2.404 | 1.397 |

Volume (m^{3}) | 5.457×10^{-4} | 5.457×10^{-4} | 2.414×10^{-4} |

Mass power density (kW/kg) | 0.355 | 0.418 | 0.458 |

Volume power density (kW/m^{3}) | 1.81×10^{3} | 1.84×10^{3} | 2.65×10^{3} |

Parameters . | Radially magnetized^{10}
. | Axially magnetized^{10}
. | Proposed machine . |
---|---|---|---|

Phase number | Three | Three | Single |

Power (kW) | 0.988 | 1.006 | 0.64 |

Current density (A/mm^{2}) | 7.701 | 7.701 | 7.8 |

Average speed (m/s) | 2.8 | 2.8 | 2.8 |

Frequency (Hz) | 100 | 100 | 100 |

Stroke (mm) | 14 | 14 | 14 |

Stator outer diameter (mm) | 100 | 100 | 100 |

Air gap length (mm) | 0.8 | 0.8 | 0.8 |

Axial length of stator (mm) | 63 | 63 | 27.5 |

Axial length of mover (mm) | 81 | 81 | 41.3 |

Stator core mass (kg) | 0.831 | 0.831 | 0.576 |

Copper mass (kg) | 0.842 | 0.842 | 0.393 |

Mover core mass (kg) | 0.491 | 0.385 | 0.055 |

PM mass (kg) | 0.622 | 0.346 | 0.373 |

Mover mass (kg) | 1.113 | 0.731 | 0.428 |

Total mass (kg) | 2.786 | 2.404 | 1.397 |

Volume (m^{3}) | 5.457×10^{-4} | 5.457×10^{-4} | 2.414×10^{-4} |

Mass power density (kW/kg) | 0.355 | 0.418 | 0.458 |

Volume power density (kW/m^{3}) | 1.81×10^{3} | 1.84×10^{3} | 2.65×10^{3} |

## V. CONCLUSION

A single-phase tubular PMLM with hybrid Halbach/axially-magnetized PM arrays is proposed and studied for the Stirling generation system. Machine topology and operating principle are elaborately described. The influence of several main structural parameters on both the force capability and power density are investigated, and the selection method for key structural parameters are presented. Compared with the conventional PMLMs, the proposed machine features high mass and volume power density, and has the advantages of simple control and low converter cost. The proposed machine topology has the advantages of both the axially-magnetized and Halbach PMLMs, and this machine topology is applicable to tubular PMLMs with any phases.

## ACKNOWLEDGMENTS

This work was supported by National Natural Science Foundation of China under Project 51325701, 51377033 and 51607046.