In this paper, an electromagnetic analysis method is proposed to improve force characteristics by using spring permanent magnets (PMs) along the axial and circumferential directions of a linear oscillatory generator (LOG). For accurate electromagnetic analysis, a detailed analysis model of the LOG is developed, and the governing equations of each subdomain are derived based on Maxwell’s equations and electromagnetic theory. Analytical solutions of the magnetic vector potential in each subdomain are derived using boundary conditions. The reliability of the proposed method is verified through a comparison with the results of a two-dimensional finite element analysis (FEA). In particular, the force characteristics depending on the spring PM is effectively derived by considering the three-dimensional (3D) circumferential and axial end effects. The proposed analysis method is used to determine the thickness of spring PM that transforms the detent force of the LOG with the spring PM into restoring force. The reliability of the proposed analysis method is verified through comparison with the results of a 3D FEA.

## I. INTRODUCTION

The Stirling-engine-based cogeneration system is one of the most effective measures for regulating greenhouse gases to cope with climate change, and this system is being commercialized mainly in major developed countries.^{1,2} In this system, the piston of a Stirling engine is connected to the permanent magnet (PM) mover of a linear oscillatory generator (LOG).^{3} In the LOG design, preventing overstroke of the PM mover is an important objective function. To prevent overstroke of PM mover, a spring PM has been used, and the detent force has been employed as a magnetic damper.^{4,5} Therefore, it is important to accurately predict and improve the force characteristics in the design process of LOGs.

In this paper, an electromagnetic analysis technique that uses the subdomain method is proposed for analyzing and improving the force characteristics of a LOG equipped with a spring PM. The subdomain method has the advantage of being faster in analysis than the finite element method (FEM) and is useful for initial and optimal design based on physical insight into the relationship between design variables and performance.^{6} In electromagnetic modeling, the actual LOG structure is converted into a simplified analytical model based on a cylindrical coordinate system by applying several assumptions.^{7} The governing equations of each subdomain are defined based on Maxwell’s equations and the constitutive equations of electromagnetism, and the general solution is derived using the variable separation method. An analytical solution can be derived using boundary conditions, and the force characteristics can be predicted by applying Maxwell stress tensor. In particular, a design technique to improve the force characteristics is presented. The reliability of the proposed method is verified through a comparison between its results and those obtained using the FEM.^{8,9} Finally, the induced voltage and output power were measured from the prototype and experimental set and compared with the proposed analysis method and FEM results. The validity of the proposed method and the reliability of the FEM results were verified through comparison with the analysis and experimental results.

## II. ELECTROMAGNETIC ANALYSIS OF PM LOG

### A. Problem description and assumptions

Figure 1(a) shows the analysis model of the LOG. The PM is connected to the moving part, and the inner and outer cores and single-phase winding are the stator parts. The PM mover reciprocates linearly, and in the initial state, the PM mover is aligned with the center of the LOG. The assumptions employed to build the simplified analytical model are described in the literature on the subdomain method.^{10} As shown in Figs. 1(b) and 1(c), the proposed analytical model is divided into nine regions: air (I, III, V, IX), PM (IV), slot opening (VI), slot (VII), and end (II, VIII) regions. The main parameters of the analytical model are defined in Table I. To impose periodic boundary conditions, the width of the periodic region is set to 2*τ*. Further, *z*_{so}, *z*_{s}, *z*_{ie}, and *z*_{oe} denote the mechanical positions of the slot opening, slot, and inner and outer ends, respectively.

Parameters . | Values (mm) . | Parameters . | Values . |
---|---|---|---|

r_{01}: Inner radius of inner core | 52.5 | τ_{so}: Width of slot opening | 4 mm |

r_{02}: Outer radius of inner core | 67.5 | τ_{sl}: Width of slot | 51 mm |

r_{03}: Inner radius of PM | 69.5 | N_{turn}: Turns per coil | 341 |

r_{04}: Outer radius of PM | 76.5 | N_{seg}: Number of stator segments | 8 |

r_{05}: Inner radius of outer core | 77.5 | N: Number of harmonics in air region | 200 |

r_{06}: Inner radius of slot | 79.5 | M: Number of harmonics in air region | 10 |

r_{07}: Outer radius of slot | 111 | L: Number of harmonics in air region | 10 |

r_{08}: Outer radius of outer core | 124 | H: Number of harmonics in air region | 10 |

τ_{ar}: Width of outer core | 77 | G: Number of harmonics in air region | 10 |

τ_{ic}: Width of inner core | 73 | x_{m}: Max stroke | 11 mm |

τ_{m}: Width of PM | 33 | Core material | 35PN230 |

τ_{s}: Width of assisted PM | 5 | PM material | N42SH |

Parameters . | Values (mm) . | Parameters . | Values . |
---|---|---|---|

r_{01}: Inner radius of inner core | 52.5 | τ_{so}: Width of slot opening | 4 mm |

r_{02}: Outer radius of inner core | 67.5 | τ_{sl}: Width of slot | 51 mm |

r_{03}: Inner radius of PM | 69.5 | N_{turn}: Turns per coil | 341 |

r_{04}: Outer radius of PM | 76.5 | N_{seg}: Number of stator segments | 8 |

r_{05}: Inner radius of outer core | 77.5 | N: Number of harmonics in air region | 200 |

r_{06}: Inner radius of slot | 79.5 | M: Number of harmonics in air region | 10 |

r_{07}: Outer radius of slot | 111 | L: Number of harmonics in air region | 10 |

r_{08}: Outer radius of outer core | 124 | H: Number of harmonics in air region | 10 |

τ_{ar}: Width of outer core | 77 | G: Number of harmonics in air region | 10 |

τ_{ic}: Width of inner core | 73 | x_{m}: Max stroke | 11 mm |

τ_{m}: Width of PM | 33 | Core material | 35PN230 |

τ_{s}: Width of assisted PM | 5 | PM material | N42SH |

### B. Magnetization modeling

^{10}The Fourier series expression for magnetization of the PM topology is as follows:

*c*

_{1}=

*r*

_{04}

*r*

_{03}/(

*r*

_{04}+

*r*

_{03}),

*c*

_{2}= 1/(

*r*

_{04}+

*r*

_{03}),

*M*

_{rcn}=

*M*

_{rn}cos(

*k*

_{n}

*z*

_{0}),

*M*

_{rsn}=

*M*

_{rn}sin(

*k*

_{n}

*z*

_{0}),

*M*

_{rn}denotes the Fourier coefficients of the main PM, and

*z*

_{0}denotes the mover position. The coefficients

*c*

_{1}and

*c*

_{2}in (1) are used to represent

*M*

_{rn}as a function of

*r*, and

*n*denotes the

*n*th-order spatial harmonics.

^{10,11}

*M*

_{rn}

^{s}denotes the Fourier coefficients of the spring PM, and

*z*

_{0s}is the distance between the centers of the main PM and spring PM.

### C. Electromagnetic field analysis of PM LOG

*μ*

_{0}denotes the permeability of vacuum.

*k*

_{n}=

*np*/

*t*,

*k*

_{h}=

*hp*/

*t*

_{ie},

*k*

_{m}=

*mp*/

*t*

_{so},

*k*

_{l}=

*lp*/

*t*

_{sl}, and

*k*

_{g}=

*gp*/

*t*

_{oe}.

*A*

_{qps},

*A*

_{qpc}, and

*A*

_{qp0}denote particular solutions.

^{11,12}The undefined coefficients (

*A*

_{0},

*B*

_{0},

*A*

_{n,h,m,l,g},

*B*

_{n,h,m,l,g},

*C*

_{n}, and

*D*

_{n}) can be determined by calculating the boundary conditions.

### D. Analytical solutions and boundary conditions

^{10–12}Furthermore, the boundary conditions of non-periodic boundaries along the

*z*-direction are calculated by applying a combination of the Neumann and continuous boundary conditions.

^{7}

**A**=

**B**, the flux density of the normal and tangential components can be expressed as

## III. CALCULATION OF ELECTROMAGNETIC FORCES CONSIDERING END EFFECT

^{12,13}

*A*

_{core,effective}and

*A*

_{core,total}denote the effective area and total area of the core.

^{12}

## IV. VALIDATION OF PROPOSED ANALYTICAL METHOD WITH FE METHOD

As shown in Figs. 4(a) and 4(b), the flux density distributions calculated using the proposed analytical method are consistent with those obtained using the nonlinear two-dimensional (2D) FEM. These results show that the proposed methods and assumptions are valid and can be used to predict electromagnetic forces.

From Maxwell stress tensor theory, the electromagnetic force is affected by the magnetic flux density in the air gap. Figure 5(a) shows the force analysis results derived from the proposed method considering axial and circumferential end effects compared with the three-dimensional (3D) FEM results. As depicted in Fig. 5(a), the force results calculated using the proposed method in the presence and absence of the spring PM are consistent with the corresponding 3D FEM results.

Figure 5(b) presents a comparison between the force characteristics values obtained in the presence and absence of the spring PM. The LOG equipped with the spring PM changes the direction of force and acts aligns it with the center of the LOG. The analysis results indicate that an analysis method that considers the circumferential and axial effects effect in the circumferential direction and the spring PM by using the principle of superposition is useful for optimizing the force characteristics in the design stage. Figure 6 shows the experimental set of LOGs equipped with a spring PM, and compared the proposed method, FEM, and experimental results. The experiment set consisted of a back-to-back system with LOGs, and position, voltage, and current were measured using a position sensor and an oscilloscope. By comparing analysis and experimental results, the validity of the proposed method and the reliability of the FEM results were verified.

## V. CONCLUSION

In this paper, an electromagnetic analysis of a LOG was performed considering the radial stacking effect of the stator, and a PM mover equipped with a spring PM was analyzed using the subdomain method. For electromagnetic analysis of the LOG, a simplified 2D analytical model and general solutions were derived for each subdomain. Undetermined coefficients of the general solutions were calculated using boundary conditions, and electromagnetic force were calculated using the derived analytical solutions. In particular, a method for calculating the force was proposed by considering stacking in the circumferential direction and considering the presence or absence of the spring PM. The validity of the proposed analytical method was verified by the results obtained using it to those obtained using the 2D and 3D FEM. The analytical method proposed herein was found to be useful used in the initial design stage and design optimization stage of the LOG, and in this method, design variables can be changed depending on the requirements and constraints of the machine structure and control system.

## ACKNOWLEDGMENTS

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1G1A1013741). This research was also supported by Korea Institute of Marine Science and Technology Promotion (KIMST) funded by the Ministry of Oceans and Fisheries (RS-2023-00254688).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Kyung-Hun Shin**: Conceptualization (lead); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). **Jang-Young Choi**: Conceptualization (equal); Funding acquisition (equal); Project administration (lead); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). **Han-Wook Cho**: Data curation (equal); Formal analysis (equal). **Min-Mo Koo**: Investigation (equal); Resources (equal). **Kyu-Seok Lee**: Validation (equal); Visualization (equal). **Sung-Ho Lee**: Conceptualization (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.