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.

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.

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 2τ. Further, zso, zs, zie, and zoe denote the mechanical positions of the slot opening, slot, and inner and outer ends, respectively.

Figure 2(a) shows the concept of magnetization modeling for the main PM by considering the axial end effect. To consider the axial end effect, a harmonic period was applied to the magnetization model described in a previous study.10 The Fourier series expression for magnetization of the PM topology is as follows:
M=c1r+c2rMrir,Mr=n=1Mrcncosknz+Mrsnsinknz,
(1)
where c1 = r04r03/(r04 + r03), c2 = 1/(r04 + r03), Mrcn = Mrn cos(knz0), Mrsn = Mrn sin(knz0), Mrn denotes the Fourier coefficients of the main PM, and z0 denotes the mover position. The coefficients c1 and c2 in (1) are used to represent Mrn as a function of r, and n denotes the nth-order spatial harmonics.10,11
Figure 2(b) shows the concept of magnetization modeling and result obtained by considering the spring PM. The magnetization model is changed as follows to consider the spring PM.
Mrcn=Mrncosknz0Mrnscosknz0+z0scosknz0z0s,Mrsn=Mrnsinknz0Mrnssinknz0+z0ssinknz0z0s,
(2)
where Mrns denotes the Fourier coefficients of the spring PM, and z0s is the distance between the centers of the main PM and spring PM.
Based on Maxwell’s equation and electromagnetic theory, the following governing equation can be obtained:
2A=μ0Jμ0×M,
(3)
where μ0 denotes the permeability of vacuum.
The governing equations of the air (I, III, V, IX) and slot opening (VI) regions, and those of the inner and outer ends (II, VIII) are presented in (4) in terms of Laplace’s equation. In addition, the governing equations of the PM (IV) and slot (VII) regions are presented in (5) in terms of Poisson’s equation.
2AθnI=0,2AθhII=0,2AθnIII=0,2AθnV=0,2AθmV I=0,2AθgV III=0,2AθnIX=0,
(4)
2AθnIV=μ0×M,2AθlV II=μ0J,
(5)
The general solution of (4) and (5) can be found using the separation of variables method. The partial differential equations of governing equations are as follows:
Aθ=n=1AnI1knr+BnK1knr+Aθnpccosknz+CnI1knr+DnK1knr+Aθnpssinknziθ,
(6)
Aθ=Aθp0+A0r+B0r+h,m,l,g=1Ah,m,l,gI1kh,m,l,gr+Bh,m,l,gK1kh,m,l,gr×coskh,m,l,gzzie,so,sl,oeiθ,
(7)
where kn = np/t, kh = hp/tie, km = mp/tso, kl = lp/tsl, and kg = gp/toe. Aqps, Aqpc, and Aqp0 denote particular solutions.11,12 The undefined coefficients (A0, B0, An,h,m,l,g, Bn,h,m,l,g, Cn, and Dn) can be determined by calculating the boundary conditions.
An analytical solution is derived by substituting the boundary conditions into the general solution. Given that there is no line current density at the boundary, the magnetic field of the tangential component and the vector potential are continuous.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 
r=r0:AθnI=0r=r1:AθnI=AθhII,BznI=BzhIIzzie,zie+τier=r2:AθhII=AθnIII,BznIII=BzhIIzzie,zie+τier=r3:AθnIII=AθnIV,BznIII=BznIV/μrIVr=r4:AθnIV=AθnV,BznIV/μrIV=BznVr=r5:AθnV=AθmVIzzso,zso+τsoAθgV IIIzzoe,zoe+τoe,BzlV=BzmVIzzso,zso+τsoBzgV IIIzzoe,zoe+τoer=r6:AθmVI=AθlV II,BzlV II=BzmVIzzso,zso+τsor=r7:BzlV II=0r=r8:AθgV III=AθnIXBznIX=BzgV IIIzzoe,zoe+τoer=r9:AθnIX=0,
(8)
Given ∇ × A = B, the flux density of the normal and tangential components can be expressed as
Br=dAθdzir,Bz=Aθr+dAθdriz.
(9)
The outer and inner cores are radial laminated to simplify manufacturing and minimize the effects of eddy currents generated in electrical steel. The difference in electromagnetic field owing to the circumferential end effect is shown in Fig. 3, and the stacking factor is calculated as follows.12,13
ksf=Acore,effectiveAcore,total,
(10)
where Acore,effective and Acore,total denote the effective area and total area of the core.
The force characteristics can be calculated using the Maxwell stress tensor. Because the LOG has a PM mover, the force is calculated considering the double air gap and stacking factor, as follows:12 
Fz=ksfπr3+r4μ0z0τsτm/2z0+τs+τm/2BrIVBzIVdz.
(11)

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.

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.

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).

The authors have no conflicts to disclose.

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).

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

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