Design and analysis of linear oscillatory single-phase permanent magnet generator for free-piston stirling engine systems

Because of increasing environmental concerns, intensive research and development have been conducted in the field of renewable energy. Among several technologies, permanent magnet linear machines (PMLMs) are an important area of study. Fixed-frequency linear motion applications, such as compressors, pumps, vibrators, and speakers/microphones use linear oscillatory single-phase permanent magnet motors (LOMs), while short-stroke applications such as FPSEs use LOG.^1–3 The efficiency of the LOG is increased by reducing the mechanical losses caused by the crankshaft mechanism. In addition, this system has lower noise and fewer vibration effects than rotary machines. In the case of the stirling engine using an LOG, the LOG has a single-phase, tubular structure, inner mover with an NdFeB PM and two air gaps to achieve low side force and high power density and efficiency.⁴ 

In this study, LOGs for FPSE systems were designed and analyzed. In order to implement the design of the LOG for suitable FPSEs, we conducted electro-magnetic analysis of the LOG with varying design parameters. The detent force was optimized using an assisted PM to utilize the mechanical strength by detent force. Then, eddy current loss analysis was conducted according to the segmented PM. Based on the analysis result, the LOG was designed to satisfy the required specifications for FPSE systems. Finally, the analysis results of the designed LOG machines were compared with the measured results.

The structure of an FPSE system is shown in Figure 1(a). The illustration shows that the FPSE system is comprised of heat exchangers, a displacer, a piston, and an LOG. The piston of the stirling engine is connected to the PM mover of the LOG. The FPSE system is driven by a stirling cycle, which consists of four thermodynamic processes. The principle behind the stirling system is that the pressure and volume increase when the gas is heated, and the pressure and volume decrease when the gas is cooled. Displacer and piston motion control this process, and the motion is dependent on the springs used. The springs are designed to move the pistons at a natural frequency equal to the engine operating frequency. The resonance frequency of FPSE systems with a mass-spring is represented below.³ 

where K is the mechanical spring constant and M is the mover weight, which includes the weight of the piston and PM. When M of the mover is reduced for the given resonance frequency, K will be smaller, reducing the cost of the spring. Therefore, the weight of the PM mover is an important design parameter. The required design specification of the FPSE system presented in this paper is shown in Table I.

Figure 1(b) shows the analysis model of an outer core type LOG for a stirling engine. Due to linear oscillating motion and short stroke, the linear generator for the stirling engine is widely used in single-phase machines. When the single-phase LOG is operated by harmonic motion, the mover position and speed can be represented using the equation below.

where x_m is the maximum value of the mover stroke and ω is the angular frequency of the oscillating motion. The electromotive force (EMF) is expressed using the equation below.

where $ψPM$ is the magnetic flux linkage of the phase coil. To obtain a sinusoidal EMF waveform, the rate of change of magnetic flux linkage must be constant with the mover position in the stroke region. Figure 2(a) and (b) show the parametric analysis result of the stroke length and EMF variation ratio with respect to the PM width w_m1 and inner radius R_mi, respectively. The other parameters of the LOG, such as the PM thickness, number of turns, air-gap length, and mover speed are constant. The illustration shows that stroke length is increased with respect to the magnet width, and the PM inner radius does not affect the stroke length. As mentioned earlier, THD of EMF is affected by the rate of change of magnetic flux linkage in the harmonic motion of the mover. Therefore, the EMF induced by a change in the magnetic flux linkage must be uniform at constant speed. As shown in figure 2(b), the EMF variation ratio rapidly increases for a PM width greater than 38 mm. From the analysis result of the stroke length and EMF variation ratio, the magnet width is determined to be 33 mm to satisfy the design specification of the FPSE system. Figure 3(a) shows the PM weight of mover according to PM width w_m1 and inner radius R_mi. Here, the PM weight of mover to be included with assisted PM as following chapter C. The design parameter of linear generator such as outer radius R_oo, inner radius R_ii, and outer core width w_s are to be selected to considering the size of the stirling engine. In particular, the mover PM weight w_m1 and w_m2 are very important parameter because the magnet weight affect resonance frequency of stirling engine. In addition, the magnet size affect to stroke and output power of linear generator. Figure 3(b) shows the parametric analysis result of the EMF with respect to the PM inner radius R_mi and PM thickness. The value of EMF increases linearly with respect to the PM inner radius R_mi, and nonlinearly with respect to the PM thickness. When the PM volume increases, the output is increased; however, the mover weight also increases. Therefore, the PM radius and thickness should be suitably selected to satisfy the design requirements. Based on the design target for the FPSE system, the PM inner radius and thickness were determined to be 69.5 mm and 7 mm, respectively.

The LOG is generated using the detent force between the PM and the iron core with respect to the mover position. The detent force due to the slot effect and end effect is a very important design parameter in linear machines. In order to use mechanical strength by detent force, the detent force was applied to the assisted PM. Here, the mechanical strength means a force that prevents the over stroke of mover PM. The generating power acts as a damping element of the stirling engine. The stirling engine may cause problems of over stroke due to breakdown of electrical load or pressure control during operation. Therefore, proper design of detent force is advantage to protect over stroke. Figure 4(a) shows the analysis result of the detent force with respect to the assisted PM width w_m2 at a minimum position stroke of -13 mm. In this illustration, only half of the assisted magnets are used in a tangential direction. The maximum stroke is approximately 20 % of rated stroke. The negative value of detent force proves the force moves in a sideways direction. Therefore, positive values of detent force are generated in order to use mechanical strength. Figure 4(b) shows the comparison between the detent force of the initial model and the assisted PM model. It shows that the direction of the detent force changed in the PM model and the force was hardly generated in the stroke region.

Although eddy current loss is usually smaller than the stator copper and iron losses, it may cause a significant rise in the temperature of the PM as the result of poor heat dissipation in the mover. In turn, this may cause irreversible demagnetization of the PM. The eddy current loss is calculated from the eddy current density as follows⁵ 

where J_e is the eddy current density in the PM, V is the volume, and σ is the electric conductivity. Figure 5(a) shows the current density distribution obtained by 3D FEA with respect to the PM segment. In order to reduce eddy current loss in the PM, the mover PM was segmented in the tangential direction. The current density of the 3-segment PM was higher than that of the 24-segment PM. Figure 5(b) shows the eddy current loss analysis result with respect to the PM segment. Based on the eddy current loss analysis result, we used the 24-segment PM in the tangential direction.

Figure 6(a) shows the photograph of manufactured model of LOG for FPSE system. In order to reduce core loss, the outer core of the LOG was manufactured using a 12-lamination block and the inner core of the LOG was manufactured with a radial laminated core. Figure 6(b) and (b) show the back-to-back test system and experimental set for evaluation of linear generator. The back-to-back test system was composed by linear actuator, linear generator, AC power supply, capacitor and load resistance. The two linear machine was connected linear coupling, and the air bearing was used for reduced friction loss. The weight of manufactured mover including two PM mover, shaft and linear coupling is 8.6 kg. The mechanical spring was used to satisfy resonance condition, and mechanical spring constant are 42 kN/m. The position of mover was measured by laser displacement sensor. When the LOG is operating in a resonance frequency, the series capacitor must be used to compensate the inductance to extract more power for a given machine at given output voltage. Therefore, the series capacitor was used for power factor correction in each the resonance frequency of test condition. Figure 7(a) shows the experimental result of LOG for no-load induced voltage using the back-to-back system. It can be seen that the experimental result is in agreement with the FEA analysis result. Figure 7(b) shows the experimental result of LOG for output power. The load experiment was conducted for a load resistance of 60 Ω and series capacitor of 100 $μF$⁠. It can be seen that the average output power is 1041 W at the frequency 30 Hz and stroke ±11 mm, and the experimental result is in agreement with the FEA analysis result. However, the experiment was not performed on the rated condition 1500 W. The reason for this lies in the increased mover weight of LOG. The back-to-back system is more required equipment such as shaft and linear coupling. Therefore, the resonance frequency is at the lower region than the designed resonance frequency. In this test condition, the maximum stroke of power of the piston is ±11 mm and the resonance frequency is 30 Hz. In order to increase the output power of LOG is to be increased in the stroke length or the operating frequency. Figure 8 show the comparison of FEA result with experimental result according to load resistance. It can be confirmed that the increase in output to be increased when the resonance frequency has reached the rated condition.

This paper presented the design and analysis of an LOG for the FPSE system. In order to implement the design of the LOG for suitable FPSEs, we conducted electro-magnetic analysis of the LOG with variable design parameters. The detent force was optimized using an assisted PM to use the mechanical strength by detent force. Then, eddy current loss analysis was conducted with respect to the segment PM. Based on the analysis result, the LOG was designed to satisfy the required design specification of the FPSE system. The initial predictions are aligned with the experimental results. The analysis and design presented in this paper will be useful in the design and analysis of the LOG for FPSE systems.