Modeling and analysis of tubular flux-switching permanent magnet linear generator with hybrid cores for stirling generator

Abstract

The hybrid magnetic core can combine the advantages of two or more magnetic materials, enhancing machine performance while reducing core loss under determined working condition. By combining soft magnetic composite (SMC) materials and silicon sheets, a tubular flux-switching permanent magnet linear generator (TFSPMLG) with a hybrid magnetic core is designed. In order to validate the advantages of the hybrid magnetic core, the performances of TFSPMLG with different kinds of cores are analyzed by finite element method (FEM). Based on the piston motion characteristics of a free-piston Stirling engine, a comparative performance analysis was conducted on the output voltage, output power, losses, and efficiency of TFSPMLG with hybrid magnetic core under different oscillation frequency of the mover. The losses calculated by FEM were employed in the thermal field analysis, and the temperature distribution of TFSPMLG under rated load operation was obtained.

Keywords

tubular flux-switching permanent magnet generator free-position stirling engine soft magnetic composite material hybrid magnetic core finite element analysis

Introduction

The free-piston Stirling engine is an externally heated engine which uses gas as the working medium. It has the advantages of high efficiency, low pollution, and flexible fuel.^1,2 The piston of the engine only performs reciprocating linear motion and its stroke is not mechanically constrained. The combination of a free-piston Stirling engine and a linear generator reduces the requirements of transmission devices such as ball screws, therefore the efficiency of the power system can be improved^3,4 and it is with the reliable self-starting capability.^5,6

The linear generator is a key component of a free-piston Stirling power generator system, which converts kinetic energy to electrical energy. In recent years, linear generators have been extensively studied. In Chen et al.,⁷ a single-phase tubular permanent magnet linear generator with high power density is proposed. In Zhu et al.,⁸ a novel double-sided flat-type permanent magnet synchronous linear generator is proposed. Because of its mover is designed with quasi-Halbach permanent magnet arrays, its torque ability and efficiency are higher than that of single-side flat type generator. In Radmanesh and Farahani,^9,10 a linear generator with modular stator and quasi-Halbach permanent magnet array mover is proposed for free-piston engines. Compared with conventional linear generators, this machine is with higher power density and efficiency.

Flux-switching permanent magnet machine (FSPMM) is a typical doubly salient permanent magnet machine, which has the advantages of simple rotor structure, high torque density and etc. The tubular flux-switching permanent magnet linear machine (TFSPMLM) is a special FSPMM, its main magnetic structure is shown in Figure 1. As shown, its stator and mover are circumferentially closed, therefore it will not be affected by transverse end effect.^11,12 As a tubular linear machine, TFSPMLM can perform linear motion without complex mechanical conversion devices like ball screws required.¹³ Since the permanent magnets of TFSPMLM are located on stator side, it is with easy heat dissipation and good anti-demagnetization ability. In TFSPMLM, the PM flux closes along the path with the minimum reluctance. As shown in Figure 2, the flux path constantly changes with the motion of the mover, resulting in variations of both amount and direction of flux linkage and back-EMF across the stator winding. When appropriate current is applied to the winding, the actuator can be driven by electromagnetic thrust. On the other hand, when the winding is connected to a load and the mover is driven by the engine, TFSPMLM will generate electrical energy as a linear generator.

Figure 1.

Main magnetic structure of TFSPMLM.

Figure 2.

Operation principle of TFSPMLM, (a) position 1, (b) position 2, (c) position 3, (d) position 4.

The magnetic core of conventional TFSPMLM is made by silicon steel sheets laminated axially,¹⁴ as shown in Figure 3(a). Though the manufacturing process of this kind of core is very simple, the alternating magnetic flux vertical to the silicon steel sheet surface may be resulted which cause a large amount of eddy current loss when the machine is working.¹⁵ The produced eddy current will reduce the machine efficiency and cause severe temperature rise in the stator part. To solve above problem, the silicon steel sheets of the magnetic core can be laminated along the circumference direction, as shown in Figure 3(b). Then, the magnetic flux can work in the silicon sheets plane. However, the stacking factor of this kind magnetic core is very low, and the manufacturing process is difficult.¹⁶ Besides the silicon sheets, soft magnetic composite (SMC) material is a relatively new soft magnetic material which can be used in designing the cores of electric machines and inductors.^17–19 Based on the molding technology, SMC can be fabricated into magnetic cores with various complex structures. Compared with silicon sheets, SMC has the merits of magnetic and thermal isotropy characteristic and low eddy current losses characteristic. However, it is also with the disadvantage of low magnetic permeability and high hysteresis loss. To overcome the disadvantages of SMC and silicon steel sheets used in designing the electrical machines, the application of a hybrid magnetic core combining silicon steel sheets and SMC is an effective way.^20–23

Figure 3.

Silicon steel sheet and SMC cores, (a) silicon sheets laminated in axial direction, (b) silicon sheets laminated in circumferential direction, (c) SMC core.

Current academic research on hybrid cores primarily concerns the combined application of non-oriented silicon steel sheets with oriented silicon steel sheets, amorphous alloys, and magnetic sheet material (SMC).^24,25 Based on the magnetic circuit of the motor, an internally magnetized permanent magnet synchronous motor featuring non-oriented silicon steel laminations for the stator yoke and oriented silicon steel laminations for the stator teeth has been designed by European researchers. Under high-load operating conditions, the motor's torque output increased by 4.7%, whilst iron losses are significantly reduced during operation below 4000 revolutions per minute.²⁶ A permanent magnet-assisted synchronous reluctance motor for electric bicycle drive applications is proposed. The stator teeth are fabricated from oriented silicon steel laminations. The stator yoke and rotor are constructed from non-oriented silicon steel laminations. Finite element analysis indicates that the motor's output torque and efficiency have increased by 9.89% and 1.7% respectively, whilst no-load iron losses have been reduced by 14.93%.²¹ A hybrid material core claw-pole motor is proposed. The stator yoke and rotor core of the motor are fabricated from SMC, while the stator claw poles are constructed from silicon steel laminations. The results demonstrate that the hybrid material core structure significantly enhances both the output torque and the no-load back-EMF amplitude of the claw-pole motor.²⁷ It can be seen that the hybrid material magnetic core has certain advantages in the machines.

In this paper, a tubular flux-switching permanent magnet linear generator (TFSPMLG) with hybrid material magnetic cores is proposed and optimized. Firstly, the main magnetic structure of TFSPMLG is introduced, its main design parameters are optimized for obtaining a good design. Secondly, the magnetic flux density distribution, permanent magnet flux linkage, no-load back-EMF and thrust force of TFSPMLGs are compared and analyzed by FEM. Thirdly, the generator performance of TFSPMLG is analyzed under resistance load. The load terminal voltage, output power, loss, and efficiency of TFSPMLG are calculated. Finally, based on the loss calculation of TFSPMLG, the steady-state temperature of the generator at rated load is obtained.

Design optimization of TFSPMLG with hybrid magnetic core

Figure 4 shows the FEM model of TFSPMLG with SMC core around the Z-axis in the 2D cylindrical coordinate system. Its main design parameters are tabulated in Table 1. The TFSPMLG with hybrid magnetic core is designed by replacing some SMC materials with silicon sheets laminated alone the tangential direction as shown in Figure 5. As shown, the silicon sheet cores are distributed on the SMC core with a determined equal angle. Specifically, two TFSPMLGs with hybrid magnetic cores are designed, named as H4 and H8, based on each stator is composed of 4 or 8 silicon sheet modules. After determining the initial design, Taguchi method is used to optimize the parameters of mover tooth of TFSPMLG with SMC core. Based on above optimization results, the key parameters W_s and H_SMC of TFSPMLG with hybrid material magnetic cores are optimized.

Figure 4.

2D structure of TFSPMLG.

Figure 5.

3D structure of TFSPMLG with hybrid material magnetic core, (a) TFSPMLG with H4 core, (b) TFSPMLG with H8 core.

Table 1.

TFSPMLG parameters.

Design parameter (uEnit)	Symbol	Value
Stator outer radius (mm)	R_so	50
Stator inner radius (mm)	R_si	25
Length of Stator (mm)	L_stator	210
Stator teeth width (mm)	W_st	3
Number of stator slots	N_s	12
PM width (mm)	W_pm	3.8
Mover outer radius (mm)	R_ro	24
Mover slot pole pitch (mm)	τ_m	15
Number of mover teeth	N_mt	14

The detent force of TFSPMLG is composed of cogging force and end force. The detent force of TFSPMLG is determined by both the cogging force and the end force, as shown in (1). The cogging force is caused by the interaction between the permanent magnet and the teeth slot structure of the machine. The end force is caused by the open ends structure of the linear motor. For reducing the end force, designing a special stator structure is an effective way.²³ The stator of TFSPMLG is composed of stator core, permanent magnets and windings, while its mover is very simple. In the optimization process, it is possible to keep the structure of the stator unchanged by keeping the mover as an optimization object. To reduce the cogging force and increase the thrust force, Taguchi method is used to optimize the parameters of mover tooth.

F_{d e t e n t} = F_{s l o t} + F_{e n d}

(1)

where F_slot is cogging force, F_end is end force.

The cogging force is mainly influenced by the slot-to-pole ratio and the structure of stator teeth. According to Wang et al.,²² the cogging force can be expressed as (2).

\begin{aligned} F_{s l o t} = \frac{π N_{m t} C}{4 μ_{0}} (R_{s i}^{2} - R_{m o}^{2}) \sum_{n}^{\infty} n G_{n} B_{r \frac{n N_{m t}}{N_{s}}} \sin (2 π n N_{m t} \frac{x}{L}) \end{aligned}

(2)

where N_s and N_mt are number of stator slots and mover teeth, L is the axial length of stator, B_r is the residual magnetic density of the PMs along the axial direction, n is the order of a harmonic, R_si and R_mo are the inner radius and outer radius of the stator, C is a constant related to the motor dimensions, G_n is Fourier decomposition coefficients, x is the linear position of the moving element, μ₀ is the permeability of free space.

When estimating end force, methods based on magnetic circuit analysis and the principle of virtual work are commonly employed.

F_{e n d} = \frac{B_{g}^{2} A e n d}{2 μ 0}

(3)

where B_g is the air-gap flux density, A_end is the effective end area.

Under the condition of keeping the pole pitch of the mover constant, W_mt, W_ms and D_ms are regarded as optimization parameters. Ultimately, a set of tooth parameters is obtained featuring high average thrust and low positioning force peaks. The optimization range and step size of design parameters are shown in Table 2. The average electromagnetic performance of the control factor at different levels can be calculated by FEM. Given the significant impact of rotor tooth parameter variations on motor performance, the optimization process employs small step adjustments for parameter tuning. The range of parameter variation is determined through a single scan.

Table 2.

Optimization variables of TFSPMLG.

Parameter (unit)	Symbol	Range	Step
Mover teeth width(mm)	W_mt	4.1–4.5	0.1
Mover teeth height(mm)	D_ms	6.6–7.8	0.3
Mover slot width (mm)	W_ms	9–9.4	0.1

To simulate all parameter combinations in Table 2, 125 computational operations are required, which will impose a considerable computational burden. Taguchi methods based on orthogonal experimental design can significantly reduce computational effort and enhance optimization outcomes through the scientific configuration of multi-factor experimental plans. Its core principle lies in implementing balanced sampling within the parameter space based on orthogonal arrays, selecting representative design points for combined analysis. The representativeness of each orthogonal test height is ensured., thereby guaranteeing the reliability of results while substantially reducing computational effort during the optimization process. The five level values corresponding to the target parameter in this orthogonal experiment are shown in Table 3.

Table 3.

Level value of target parameters.

Control factor	W_mt (mm)	D_mst (mm)	W_mst (mm)
Level 1	4.1	6.6	9
Level 2	4.2	6.9	9.1
Level 3	4.3	7.2	9.2
Level 4	4.4	7.5	9.3
Level 5	4.5	7.8	9.4

The mean values of the performance parameters, as well as the mean values of the performance parameters at different levels of each control factor, are obtained from (4) and (5) respectively.

m = \frac{1}{n} \sum_{i = 1}^{n} m_{i}

(4)

m = \frac{1}{l e v e l} \sum_{i = 1}^{l e v e l} F

(5)

The electromagnetic performance of the control factor at different levels can be calculated through FEM as shown in Figure 6. With the increase of W_mt and D_ms, the peak-to-peak value of detent force increases gradually. W_mt has a significant influence on detent force. The average value of thrust force shows the maximum value with the increase of all three control factors. The combination level of the initially selected control factors is (1,2,4). It was found that the result of W_mt, D_ms and W_ms are 4.1, 7 and 9.3 mm respectively, the machine is able to significantly reduce the detent force while increase the thrust force.

Figure 6.

Average electromagnetic performance of each control factor at different levels, (a) peak-to-peak value of detent force, (b) average value of thrust force.

To solve the problem of low magnetic permeability and high hysteresis loss in SMC cores, part of the SMC material of TFSPMLG is replaced by the silicon sheets stacked in tangential direction. Figure 7 shows the axial sections of H4 and H8. As shown, the adopted silicon sheet module is determined by the tangential stacking width of W_s and the height from bottom of the stator inner radius to the silicon sheet cores of H_SMC. The minimum values of W_s and H_SMC are both equals to 0, and their maximum values can be calculated according to (6) and (7), respectively, where R_si is the inner radius of the stator, and n is the number of silicon steel modules in the machine.

W_{s \max} = 2 * (R_{s i} + H_{S M C}) * \tan \frac{π}{n}

(6)

H_{S M C max} = R_{s o} - R_{s i}

(7)

Figure 7.

Axial section of H4 core and H8 core.

To obtain the influence of W_s and H_SMC on the main magnetic performance of TFSPMLG, FEM is used. The average thrust force is determined as the optimization objective. The influence of W_s and H_SMC on the average thrust force of H4 and H8 is shown in Figure 8. When W_s is 45 mm and H_SMC is 3 mm, the average thrust force of the H4 reaches its maximum value, which is 652.6 N, when W_s is 19 mm and H_SMC is 2 mm, the average thrust force of the H8 reaches its maximum value, which is 657.1 N. With the adoption of hybrid material core, the main working magnetic fluxes of the TFSPMLG are not vertical to the silicon steel sheet plane and the total magnetic reluctance of stator is lower than that with SMC cores. Therefore, the main magnetic performance of this machine is better than that with only SMC cores or silicon steel sheet cores.

Figure 8.

Influence of W_s and H_SMC on the average thrust force of TFSPMLG with H4 and H8, (a) H4 core, (b) H8 core.

Magnetic performance analysis of TFSPMLG

After the main magnetic structure and design parameters of TFSPMLG with hybrid magnetic cores are determined, one-eighth model of H4 core and one-sixteen model of H8 core of TFSPMLG are established for the further electromagnetic performance analysis. Based on the FEM, the comparative analyses of them are carried out in this section.

No load magnetic flux density distribution

Figure 9 shows the magnetic flux density distribution of the H4 and H8. As shown in Figure 9(a) and (c), the magnetic flux density distributed in the SMC core especially at the end part of the stator teeth is uniform. As shown, the magnetic flux density distributed at the splicing surface of the SMC and the silicon steel module is not significantly changed, which indicates that the magnetic flux is smooth between these two magnetic materials. However, the magnetic flux changes significantly between these two magnetic materials along the tangential direction due to the magnetic flux is difficult to pass through the silicon steel sheets in the tangential direction. As shown in Figure 9(b) and (d), due to the isotropy characteristic of SMC, the magnetic flux in the SMC section exhibits a divergent distribution.

Figure 9.

Magnetic density distribution of TFSPMLM, (a) magnetic density distribution of H4, (b) magnetic density vector of H4, (c) magnetic density distribution of H8, (d) magnetic density vector of H8.

Limited by the stacking direction of silicon steel sheets, the magnetic flux in the silicon steel sheets will flow in the plane of the laminations. The magnetic flux in the air gap will first pass through the SMC before entering the silicon steel module, which helps to reduce the component of magnetic flux along the stacking direction of silicon steel sheets. The eddy current loss caused by magnetic flux passing vertically through the silicon steel stack is reduced eventually. Comparing H4 and H8, it can be seen that the more modules employed in the core, the lower magnetic flux component can be achieved along the silicon sheet stacking direction.

Taking silicon steel module of H8 as a reference. The magnetic flux density in the y-direction of some determined points at the splicing surface and center surface inside the silicon steel module are calculated, the selected points are shown in Figure 10. As shown, the magnetic flux density in the y-direction is the direction vertical to the surface of silicon sheets.

Figure 10.

Distribution of sampling points for silicon sheet core.

Figure 11 shows the calculated magnetic flux density of the picked points in one electrical period. As shown in Figure 10, point from 1 to 6 are close to the radial surface between SMC module and silicon steel module. The calculated magnetic flux density of these points along the y-direction is shown in Figure 11(a). As shown, the tangential magnetic density of the point which is close to the central surface is lower than that far away from the central surface. The tangential magnetic density of the point which is at the inner part is lower than that at the radial surface. The main reason is that the magnetic flux distributed divergently in SMC part while the direction of magnetic flux is constrained effectively by the silicon steel module. In general, the tangential magnetic density inside the silicon steel module is very small, which indicates that the hybrid magnetic core structure can effectively reduce the tangential component of the magnetic density inside the silicon steel module and avoid generating significant eddy current loss.

Figure 11.

Y-axis component of magnetic density at the sampling points, (a) 1–6, (b) 6–13, (c) 13–18, (d) 18–25.

PM flux linkage and back-EMF comparison

The PM flux linkage of the TFSPMLG can be expressed as (8), where ψ_max is the maximum value of the PM flux linkage which can be expressed as (9), where N is the number of winding turns, and Φ_max is the maximum value of the magnetic flux.

ψ_{m} = ψ_{max} \cos (2 π \frac{x}{τ_{m}})

(8)

ψ_{max} = N Φ_{max}

(9)

Figure 12(a) shows the B-phase PM flux linkage waveforms of TFSPMLG with SMC, H4 and H8 core. It can be seen that the PM flux linkage waveforms of the three TFSPMLGs are basically coincident and have good sinusoidal characteristics. Since the PM flux linkage is generated by the permanent magnet, and the hybrid core does not obviously change the reluctance in the magnetic circuit. Therefore, there is no significant difference in the magnitude of flux linkage between the three kinds of TFSPMLGs.

Figure 12.

Permanent magnet flux linkage and no-load back-EMF of H4, H8 and SMC cores, (a) permanent magnet flux linkage, (b) no-load back-EMF.

As shown in (10), The no-load back-EMF of TFSPMLG can be calculated by differentiating the PM flux linkage. Figure 12(b) shows the no-load back-EMF waveforms of these three machines under the speed of 2 m/s. Due to the similar magnitude of the PM flux linkage of these machines, their amplitude and frequency of the no-load back-EMFs are basically the same.

E = \frac{d ψ_{m}}{d t}

(10)

where E is the induced electromotive force of the motor.

Inductance comparison

Figure 13 shows the B-phase winding inductance waveforms of the TFSPMLG with SMC core, H4 and H8. The average B-phase winding inductance of above machines are 10.76, 12.2, 12.06 mH, respectively. Due to the addition of silicon sheet modules in the hybrid material stator core, the magnetic permeability of the TFSPMLG with hybrid core has slightly increased, therefore H4 and H8 cores have higher inductance.

Figure 13.

Induction of H4, H8 and SMC cores.

Detent force and thrust force comparison

The detent force of TFSPMLG is determined by both the cogging force and the end force, as shown in (1). According to (2), the adoption of the hybrid core does not change the slot-to-pole ratio and tooth structure of the mover. The only difference is the magnetic permeability of the stator core. Although the hybrid core structure has an influence on the detent force, the influence is relatively small. The detent force waveforms of SMC core, H4 and H8 obtained through FEM are shown in Figure 14(a). According to the calculation results, the difference in detent force among the three types of hybrid magnetic cores is small. For the amplitude of detent force, TFSPMLG with SMC core is 53.4 N, H4 is 56.6N, and H8 is 53 N.

Figure 14.

Detent force and thrust force of H4, H8 and SMC cores, (a) detent force, (b) thrust force.

The thrust force is also an important performance for evaluating the performance of a linear motor. The average thrust force shows the output capability of the motor. The force ripple determines whether the motor can run smoothly. Figure 14(b) shows the thrust force waveforms of TFSPMLG with SMC core, H4, and H8 under sinusoidal current excitation with a rated current density of 6 A/mm². It can be seen that the thrust force waveforms of the three machines are basically coincident. Average thrust force of the TFSPMM with SMC core is 631.8 N, the force ripple is 10.1%. The average thrust force of the H4 is 652.6 N, and the force ripple is 11%. The average thrust force of the H8 is 657.2 N, and the force ripple is 9.7%. Based on the simulation results of the detent force and thrust force of the TFSPMLG, the thrust force of H4 and H8 is higher than that with SMC core. Moreover, TFSPMLG with H8 core has the best performance.

Load performance of TFSPMLG

According to the operation principle of FSPMM, its PM flux linkage across the stator winding changes periodically with its rotor rotates. When the winding is connected to a load and the mover is driven by an engine, TFSPMLG will generate electrical energy. Based on the previous comparison results, TFSPMLG with H8 core has the best performance. The generator performance of TFSPMLG under the condition of resistance load is analyzed in this section.

For the traditional electrical machine, its air magnetic flux is generally designed with the sinusoidal wave, and the produced no load back EMF is with the sinusoidal waveform as well. According to (11), it can be seen that the back-EMF of a tubular linear generator in a uniform motion state is sinusoidal with constant frequency and amplitude.

E = V \frac{d ψ_{m}}{d x}

(11)

where V is synchronous speed of linear motors, x is the displacement of the motor's armature, where ψ_m donates the same PM flux linkage of the TFSPMLG as defined in equation (8).

For an actual linear generator, its mover cannot be moved along one direction with a constant speed, due to the limitation of the mover length. In most cases, linear generators are connected with oscillate primer to generate electricity energy, such as a free-piston Stirling engine. According to the experimental results in Zare and Tavakolpour-Saleh,²⁸ it can be concluded that the displacement and velocity of the piston in a free-piston Stirling engine are approximately sinusoidal and cosine curves, as shown in Figure 15. In this section, the motion of the free-piston Stirling engine is used as the speed of the mover while analyzing the performance of TFSPMLG.

Figure 15.

Piston motion of free-piston stirling engine, (a) velocity, (b) displacement.

When the speed of the mover is sinusoidal, the back-EMF is an oscillating function of variable amplitude and variable frequency. Under the resistance load of 40 Ω and an oscillation frequency of 20 Hz, the calculated output voltage and output power waveforms are shown in Figure 16.

Figure 16.

Load terminal voltage and output power during oscillation power generation, (a) load terminal voltage, (b) output power.

Given the stroke of TFSPMLG mover is 90 mm. Three-phase star-connected resistance is connected to the external circuit of the generator. The performances of the TFSPMLG with H8 core at oscillation frequency of 10, 20 and 30 Hz have been calculated as shown in Figure 17. As shown, when the stroke of the mover is determined, the faster the oscillation, the higher the output voltage and power. The voltage adjustment rate of TFSPMLG is relatively high. The output power of TFSPMLG reaches its maximum value as the resistance increases, and then continuously decreases.

Figure 17.

Performance of TFSPMLG with different load and oscillation frequency, (a) load terminal voltage, (b) output power, (c) core loss, (d) copper loss, (e) pm loss, (f) efficiency.

For the core loss analysis, the Bertotti core loss separation formula is adopted which is expressed as (12),

P_{F e} = P_{h} + P_{ε} = k_{h} f B_{m}^{2} + k_{ε} f^{2} B_{m}^{2}

(12)

where P_h is hysteresis loss, and P_ε is eddy current loss. k_h and k_ε are hysteresis loss coefﬁcient and eddy current loss coefﬁcient respectively, f is the frequency and B_m is the magnitude of magnetic ﬂux density. However, the formula (12) is only applicable to the hysteresis loss calculation of magnetic fields without DC bias. According to the principle of hysteresis loss, the PMs of TFSPMLG will cause DC bias in the stator core, especially in the silicon steel sheets module. The hysteresis loss when considering the effect of DC bias is calculated as (13).²⁹

P_{h d} = (0.65 B_{d c}^{2.1} + 1) P_{h}

(13)

where P_hd is the hysteresis loss of DC bias, B_dc is the magnitude of DC bias. Figure 17(c) shows the calculated core loss of TFSPMLG under different working frequency and loads. With the increase of the oscillation frequency and load resistance, the loss of TFSPMLG increases. As shown in Figure 17(d), as the load resistance gradually increases, the copper loss of the TFSPMLG gradually decreases due to the decrease of armature current. Moreover, as shown in Figure 17(e), the eddy current loss generated by the permanent magnets increases with the increase of the oscillation frequency and load resistance.

When the output power and losses are obtained, the efficiency of the TFSPMLG can be calculated by (15). As shown in Figure 17(f), the efficiency of the TFSPMLG is higher than 90% in a certain load range, and the maximum operating efficiency is about 95%.

η = \frac{P_{o u t}}{P_{i n}}

(14)

P_{i n} = P_{o u t} + P_{F e} + P_{C u} + P_{P M}

(15)

where η is the efficiency of the generator, P_in is the input power of the generator, P_out is the output power of the generator, P_Fe is the iron loss of the generator, P_Cu is the winding copper loss of the generator, P_PM is the eddy current losses in permanent magnets.

Figure 18 shows the performance comparison between the TFSPMLG with H8 and SMC core. As shown, TFSPMLG with H8 core has a higher terminal voltage, higher output power, and lower core loss than that with SMC core. Moreover, with the load decreases, these difference increases.

Figure 18.

Performance comparation of TFSPMLG with H8 and SMC cores, (a) load terminal voltage, (b) output power, (c) core loss.

The following conclusions can be drawn from the analysis of the performance of TFSPMLG with H8.

The output terminal voltage of TFSPMLG is an oscillating waveform with variable amplitude and variable frequency, therefore the output terminal voltage and output power fluctuate greatly. It is necessary to cooperate with rectifier and filter to generate electrical power stably in practical applications.

TFSPMLG has high power generation efficiency in a certain range. However, TFSPMLG shows high voltage regulation when connecting to the load.

The output power of the TFSPMLG increases with the oscillation frequency increases. Compared with the SMC core, H8 core has better performances and less loss, verifying the advantages of the hybrid material core structure.

Thermal field analysis of TFSPMLG

Excessive temperature rise can lead to deterioration of insulation, demagnetization of the permanent magnets and other undesirable effects of TFSPMLG. And the thermal field analysis is very important for the electrical machine design.

According to the heat transfer theory, heat is transferred through three ways: thermal conduction, thermal convection, and thermal radiation. Since the TFSPMLG is a low-temperature heat source, the effect of thermal radiation on the machine temperature is very small, which can be ignored in the calculation.

Calculation of thermal parameters

Thermal conduction is the most important way to transfer heat for electrical machine. TFSPMLG with hybrid cores has a relatively complex structure with many splicing surfaces between the different cores. Therefore, determining the thermal conductivity between different material is very important. The thermal conductivity of the materials used in the developed TFSPMLG is shown in Table 4.

Table 4.

Thermal conductivity.

Material	Thermal conductivity (W/m °C)
M19_29G	40
NdFe30	10
SMC	21
Copper	387.6
Air	0.0267

The heat convection coefficient can be obtained directly from relational formulas based on fluid dynamics and heat transfer theory. The heat convection coefficient on the outer surface of the stator of TFSPMLG under natural air flow can be expressed as (16)

α_{s} = α_{0} (1 + k \sqrt{V})

(16)

where α_s is the heat convection coefficient on the outer surface of the stator, α₀ is the heat convection coefficient of an object in stationary air, with a magnitude of 14.2, k is the blowing efficiency coefficient, with a value of 1.3.

Considering the mover of the TFSPMLG is an ideal cylinder, according to the heat transfer principle, the heat convection coefficient between the mover surface and the air gap can be expressed as (17).

α_{m} = \frac{N_{u} λ}{D}

(17)

where α_m is the heat convection coefficient between the surface of the actuator and the air gap. λ is the thermal conductivity of air. D is the outer diameter of the mover. N_u is the Nusselt number, which can be defined as (18)

N_{u} = 0.3 {Re}^{0.6} \Pr^{0.36}

(18)

where Re is the Reynolds number and Pr is the Prandtl number of air, which can be respectively expressed as (19) and (20).

Re = \frac{ρ V D}{μ}

(19)

Pr = \frac{μ c_{p}}{λ}

(20)

where ρ is the density of air, V is the speed of the air, c_p is the specific heat capacity of air, μ is the viscosity of the air.

The airflow distribution on the mover end surface is similar to the airflow crossing over the special shape objects.³⁰ The Nusselt number of the mover end airflow can be expressed by (21) and the heat convection coefficient of the TFSPMLG mover end surface can be calculated from (17)

N_{u} = 0.43 + 0.53 {Re}^{1 / 2} \overset{1 / 3}{Pr}

(21)

Thermal Field analysis

In the thermal field analysis, the magnetic-field coupling method generally includes the one-way coupling method and double way coupling method, as shown in Figure 19. For the one-way coupling method, after each loss, the thermal conductivity of employed material and the heat convection coefficient of each surface are obtained then the thermal rise can be calculated. In the analyzed operating regime, electromagnetic losses dominantly influence the thermal field, while the feedback effect of temperature on electromagnetic properties is secondary. This approach provides significant advantages for extensive parametric optimization studies, reducing solution time compared to double way coupled methods while maintaining engineering accuracy.

Figure 19.

Magnetic-thermal coupling analysis diagram, (a) one-way coupling, (b) cyclic coupling.

The temperature distribution of the TFSPMLG under the rated load of 20 Ω resistance and the mover oscillation frequency of 30 Hz is calculated, and the obtained steady-state temperature rise distribution are shown in Figure 20. It can be seen that the temperature at the end of the TFSPMLG is lower than that in the middle, since the magnetic circuit at the end is incomplete and the heat generation rate is low. Moreover, the surface area of the end part is larger, leading to a better heat dissipation. When TFSPMLG operates under the rated load, the armature current is small and produces less copper loss. The maximum temperature of the motor is 41 °C. Comparing the temperature rise of TFSPMLG with H8 and SMC core, their steady-state temperatures rise are close.

Figure 20.

Steady-state temperature distribution of TFSPMLG with H8 and SMC cores, (a) H8, (b) SMC.

Conclusion

In this paper, a comparative study of TFSPMLGs with SMC core and hybrid magnetic cores were carried out. Two TFSPMLGs with hybrid magnetic cores, namely H4 and H8, were designed and analyzed. The following conclusions can be drawn:

The performance of TFSPMLMG with H4 and H8 are significantly influence by the width and height of silicon sheet core module. In this machine, the tangential magnetic flux density inside the silicon sheet core module is very small and will not generate significant eddy current losses. The thrust force of TFSPMLMG with H4 and H8 cores has been improved about 20.8 N and 25.3 N if compared with that of SMC core, respectively.

The output voltage and output power of TFSPMLG increase with the oscillation frequency increases, and the maximum operating efficiency can reach to about 95%. TFSPMLMG with H8 core has better power generation performance and lower loss than that with SMC cores. When it operates at rated load and speed, the steady-state temperature is 41 °C. It is proved that continuous operation of the TFSPMLG will not cause damage to the winding or demagnetization of the permanent magnets.

Footnotes

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant 52377006 and 52007047, in part by the S&T Program of Hebei Province of China under Grant 225676163GH.

ORCID iDs

Chengcheng Liu

Youhua Wang

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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