Electromagnetic Loss Analysis of a Linear Motor System Designed for a Free-Piston Engine Generator

: A free-piston engine generator is a new type of power generating device, which has the advantages of high e ﬃ ciency and simple structure. In this paper, a linear motor system composed of a moving-coil linear motor with axial magnetized magnets and a H-bridge pulse-width modulation (PWM) rectiﬁer is designed for portable free-piston engine generators. Based on the ﬁnite-element model of the motor and physical model of the rectiﬁer, the combined electromagnetic model is presented and then validated by the prototype-tested results. The electromagnetic processes of the linear motor system are simulated. The electromagnetic losses during the standard working cycle are analyzed. Under the rated reciprocating frequency of 50 Hz and the rated reciprocating stroke of 36 mm, the mechanical-to-electrical energy conversion e ﬃ ciency of 86.3% can be obtained by the linear motor system, which meets the requirement of portable free-piston engine generators. of


Introduction
A free-piston engine generator (FPEG) is a novel power generating device, which is a combination of a free-piston engine and a linear motor [1]. This new type of power generating device has attracted more and more scholars' attention due to its potential advantages of high efficiency, simple structure and renewable alternative fuels adaptability [2]. The device can be used in portable power generation, electric vehicles and communication base stations [3].
The structure of a portable FPEG, studied in this paper, is shown in Figure 1. The device consists of a mechanical body, a rectifier for power conversion, a battery for energy storage and a controller. The mechanical body includes a free-piston engine, an air-spring for kickback, and a linear motor. Compared with the traditional engine generator, the crankshaft is canceled, and the free-piston is directly connected to the mover of a linear motor. The piston-mover assembly is the only moving part of the device [4]. This structure has the advantages of compact, low mass, easy manufacture, and low maintenance cost.  The linear motor and the rectifier make up the linear motor system of the FPEG. Its performance affects the energy utilization efficiency of the FPEG. Many researchers have been working on the design of the linear motor for an FPEG [5][6][7]. Wang [8] proposed and designed a three-phase movingmagnet linear motor for the application of an FPEG. Its electromagnetic characteristics were predicted by analytical expressions and verified by finite element methods.
Xu [9] proposed a single-phase moving-coil linear motor with radial magnetized magnets for the application of an FPEG, its analytical model was created, and its thrust force and the coil inductance were investigated. This kind of single-phase moving coil linear motor has the advantage of less moving mass, faster response and better controllability, but the generating efficiency is not high enough as there is a high eddy loss in the radial magnetized magnets.
Chen [10] proposed a single-phase, short-stroke and large thrust axial magnetized linear motor. A combined electromagnetic model composed of a finite element model and mathematical model of As the free-piston is not limited by the crankshaft, the reciprocating motion of the piston-mover assembly completely depends on the instantaneous forces acted on it. The forces include combustion pressure, electromagnetic force and spring force. The compression ratio of a FPEG is variable, thus multiple renewable alternative fuels can be used by the device without any structural changes. These features make the development of a FPEG competitive and valuable.
A working cycle of the portable FPEG, studied in this paper, is shown in Figure 2. A working cycle has two strokes: compression stroke and expansion stroke. In the compression stroke, the piston-mover assembly moves up from the bottom turning center (BTC) to the top turning center (TTC), and the mixture of air and fuel in the combustion chamber is compressed. In the expansion stroke, the compressed mixture of air and fuel is burned and releases heat instantly, so the piston-mover assembly moves down from the TTC to BTC driving by the combustion pressure.
Electronics 2020, 9, x FOR PEER REVIEW 2 of 11 piston-mover assembly moves up from the bottom turning center (BTC) to the top turning center (TTC), and the mixture of air and fuel in the combustion chamber is compressed. In the expansion stroke, the compressed mixture of air and fuel is burned and releases heat instantly, so the pistonmover assembly moves down from the TTC to BTC driving by the combustion pressure. During the stable running, the linear motor works in generating mode all the time. The electromotive force (EMF) in the coil of the motor is quasi-sine wave, and the generating current in the coil is controlled as standard sine wave by the rectifier, and the electrical energy will be stored in the battery in the form of direct current. During the starting of the device, the linear motor must work briefly in motoring mode to start the motion of the piston-mover assembly.  The linear motor and the rectifier make up the linear motor system of the FPEG. Its performance affects the energy utilization efficiency of the FPEG. Many researchers have been working on the design of the linear motor for an FPEG [5][6][7]. Wang [8] proposed and designed a three-phase movingmagnet linear motor for the application of an FPEG. Its electromagnetic characteristics were predicted by analytical expressions and verified by finite element methods.
Xu [9] proposed a single-phase moving-coil linear motor with radial magnetized magnets for the application of an FPEG, its analytical model was created, and its thrust force and the coil inductance were investigated. This kind of single-phase moving coil linear motor has the advantage of less moving mass, faster response and better controllability, but the generating efficiency is not high enough as there is a high eddy loss in the radial magnetized magnets.
Chen [10] proposed a single-phase, short-stroke and large thrust axial magnetized linear motor. A combined electromagnetic model composed of a finite element model and mathematical model of During the stable running, the linear motor works in generating mode all the time. The electromotive force (EMF) in the coil of the motor is quasi-sine wave, and the generating current in the coil is controlled as standard sine wave by the rectifier, and the electrical energy will be stored in the battery in the form of direct current. During the starting of the device, the linear motor must work briefly in motoring mode to start the motion of the piston-mover assembly.
The linear motor and the rectifier make up the linear motor system of the FPEG. Its performance affects the energy utilization efficiency of the FPEG. Many researchers have been working on the design of the linear motor for an FPEG [5][6][7]. Wang [8] proposed and designed a three-phase moving-magnet Electronics 2020, 9, 621 3 of 11 linear motor for the application of an FPEG. Its electromagnetic characteristics were predicted by analytical expressions and verified by finite element methods.
Xu [9] proposed a single-phase moving-coil linear motor with radial magnetized magnets for the application of an FPEG, its analytical model was created, and its thrust force and the coil inductance were investigated. This kind of single-phase moving coil linear motor has the advantage of less moving mass, faster response and better controllability, but the generating efficiency is not high enough as there is a high eddy loss in the radial magnetized magnets.
Chen [10] proposed a single-phase, short-stroke and large thrust axial magnetized linear motor. A combined electromagnetic model composed of a finite element model and mathematical model of the controller was built, and the simulated results were compared with the tested results. This kind of single-phase axial magnetized linear motor was reported to have higher efficiency than radial magnetized. In this research, the electromagnetic loss in the power converter was not considered.
Zheng [11] proposed a 1 kW moving-magnet linear motor for the FPEG application. The power densities, thrust fluctuation, and thermal field of the linear motor were investigated, but the electromagnetic efficiency of the linear motor was not reported in the paper. Xu [12] designed a similar 3 kW linear motor with I-shaped and rectangular designs for the same application. The theoretical efficiency of 90% to 95% was reported under the reciprocating moving speed of 3 to 5 m/s.
In reference [13], a single-phase moving-magnet linear motor was proposed for an FPEG. The three-dimensional finite element model was established, and its boundary conditions were presented and explained in detail. By using the model, an electromagnetic analysis was carried out. The performance of the linear motor under reciprocating frequency 75 Hz was investigated and analyzed. The electromagnetic efficiency of 82% was obtained by the linear motor.
As per the literature review above, some informative works about the linear motor designed for a FPEG have been reported. Analytical and finite element models have been created to analyze the efficiency of the linear motor, but few works try to research the linear motor and its power driving system as a whole system. The authors believe that a combined electromagnetic loss analysis is an efficient path to assess and optimize the total efficiency of a linear motor system [14,15].
In order to meet the requirements of a portable FPEG, this paper will present a novel linear motor system, which is considered to have the advantages of low moving mass, fast response, easy controllability and high efficiency. Based on the finite-element model of the motor and physical model of its power driving system, a combined electromagnetic model is presented and validated by the prototype. Then the electromagnetic loss of the motor system is analyzed in detail.

Linear Motor Design
The linear motor system is the most important component of the FPEG. It must meet some special requirements of the FPEG, such as low moving mass, fast response, easy controllability and high energy conversion efficiency [16,17]. A single-phase moving-coil linear motor with embedded axial magnetized magnets is designed in this paper. As shown in Figure 3, the motor can be divided into three parts: the stator, the mover and the shell. The stator is made up of an inner stator and an outer stator, each stator includes an axial magnetized magnet and two iron cores, and the magnet is located between the two iron cores. The magnetization direction of magnet in the inner and outer stators is opposite. The mover is made up of two reverse series moving-coils and a high strength plastic support. The mover is located in the air gap between the inner and outer stators. into three parts: the stator, the mover and the shell. The stator is made up of an inner stator and an outer stator, each stator includes an axial magnetized magnet and two iron cores, and the magnet is located between the two iron cores. The magnetization direction of magnet in the inner and outer stators is opposite. The mover is made up of two reverse series moving-coils and a high strength plastic support. The mover is located in the air gap between the inner and outer stators.   Table 1 lists the main parameters of the linear motor in detail. Additionally, the neodymium iron boron "N42SH" is a choice for the magnet material to achieve higher residual magnetic flux density, temperature stability and coercive force. The electrical pure iron "DT4" is a choice for the core material. Compared with silicon steel sheets and low carbon steel, the electrical pure iron is more suitable for this kind of moving-coil linear motor. It has lower manufacturing costs and better permeability performance. A kind of aluminum alloy is a choice for the shell material for low density and low magnetic permeability. A kind of high strength plastic "PTFE" is a choice for the coil support material. It has the advantages of non-magnetic, non-conductive and high strength.

Rectifier Design
The rectifier used in the FPEG not only must have high efficiency, but also must have two additional functions. The first function is that the rectifier must be controllable for adjusting the generating current of the motor. The second function is that the rectifier must be used as an inverter during the starting process of the FPEG to start the motion of the piston-mover assembly. This is because the reciprocating motion of the piston-mover assembly in a FPEG completely depends on the instantaneous forces acted on it. Between the forces of combustion pressure, electromagnetic force and spring force, the electromagnetic force is the only one which can be controlled to adjust the piston motion. The control of electromagnetic force of the motor is to control the coil current of the motor according to every stroke, so the rectifier must be controllable.
An H-bridge pulse-width modulation (PWM) rectifier is designed to meet the requirements of the FPEG. As shown in Figure 4, this is a voltage source PWM rectifier. During the stable running of the FPEG, the linear motor works in generating mode all the time, and the rectifier works in rectify and boot mode. The electromotive force (EMF) in the coil of the motor is quasi-sine ware. When the EMF of the motor is positive, K3 will be controlled by PWM signals and the others remain off, and its Electronics 2020, 9, 621 5 of 11 current path of PWM on and PWM off is presented in Figure 4. When the EMF is negative, K1 will be controlled by PWM signals and the others remain off. During the starting process of the FPEG, the linear motor can briefly work in motoring mode. At this point, the PWM rectifier works as a PWM inverter. When the linear motor needs to be motoring from bottom to up, K1 keeps on, K4 will be controlled by PWM signals, and the others remain off. When the linear motor needs to be motoring from up to bottom, K3 keeps on, K2 will be controlled by PWM signals, and the others remain off. We call this type of control the limited unipolar control method. This control method can reduce the electromagnetic loss as much as possible. Table 2 lists the main parameters of the designed PWM rectifier in detail, including the control parameters and electromagnetic parameters of the power switch devices. As shown in the table, metal oxide semiconductor field effect transistors (MOSFET) are a choice for the power switch to reduce the electromagnetic loss. The parameters will be used by the model in the next section.

Linear Motor Model
The electromagnetic model of linear motors can be divided into an analytical model and finite element model. Compared to the analytical model, the finite element model has the advantages of being easy to model and quick to calculate. Additionally, the finite element model can consider the end effect and take magnetic flux leakage into account and calculate more accurately. In this paper, a transient finite element method with a motion solver is used to model the linear motor designed for the FPEG. During the starting process of the FPEG, the linear motor can briefly work in motoring mode. At this point, the PWM rectifier works as a PWM inverter. When the linear motor needs to be motoring from bottom to up, K1 keeps on, K4 will be controlled by PWM signals, and the others remain off. When the linear motor needs to be motoring from up to bottom, K3 keeps on, K2 will be controlled by PWM signals, and the others remain off. We call this type of control the limited unipolar control method. This control method can reduce the electromagnetic loss as much as possible. Table 2 lists the main parameters of the designed PWM rectifier in detail, including the control parameters and electromagnetic parameters of the power switch devices. As shown in the table, metal oxide semiconductor field effect transistors (MOSFET) are a choice for the power switch to reduce the electromagnetic loss. The parameters will be used by the model in the next section.

Linear Motor Model
The electromagnetic model of linear motors can be divided into an analytical model and finite element model. Compared to the analytical model, the finite element model has the advantages of being easy to model and quick to calculate. Additionally, the finite element model can consider the end effect and take magnetic flux leakage into account and calculate more accurately. In this paper, a transient finite element method with a motion solver is used to model the linear motor designed for the FPEG. The model of the linear motor is shown in Figure 5. This model is a two-dimensional transient finite element model with a motion region. At this point, the mover is in the middle of the stroke, and the position is defined as the original point of the mover, as shown in Figure 5. The magnetic field is mainly distributed in ferromagnetic materials, and the leakage flux is relatively small. In order to display the internal structure, the meshes in the air box of the calculation region are hidden.
Electronics 2020, 9, x FOR PEER REVIEW 6 of 11 The model of the linear motor is shown in Figure 5. This model is a two-dimensional transient finite element model with a motion region. At this point, the mover is in the middle of the stroke, and the position is defined as the original point of the mover, as shown in Figure 5. The magnetic field is mainly distributed in ferromagnetic materials, and the leakage flux is relatively small. In order to display the internal structure, the meshes in the air box of the calculation region are hidden.

Combined Simulation Model
In order to analyze the electromagnetic loss of the designed PWM rectifier, its physical model is established by using the power electronics toolbox in the software of MATLAB. Supported by the finite element software we used to create the model of the linear motor, the finite element model of the linear motor is embedded into the MATLAB software. The MATLAB software calls the finite element software in every time steps, so the real combined simulation can be realized.
As shown in Figure 6, the combined model consists of four parts: linear motor finite element model, rectifier model, rectifier controller, and free-piston engine model. The rectifier controller provides PWM signals to the four power switch devices of the rectifier model. The PWM signals are calculated based on the current feedback of the motor, and a discrete proportional integral differential (PID) algorithm is used in this paper. The free-piston engine model is created according to the difference equations described in [16]. The free-piston engine model is used to output the piston motion trajectory to drive the linear motor finite element model.

Combined Simulation Model
In order to analyze the electromagnetic loss of the designed PWM rectifier, its physical model is established by using the power electronics toolbox in the software of MATLAB. Supported by the finite element software we used to create the model of the linear motor, the finite element model of the linear motor is embedded into the MATLAB software. The MATLAB software calls the finite element software in every time steps, so the real combined simulation can be realized.
As shown in Figure 6, the combined model consists of four parts: linear motor finite element model, rectifier model, rectifier controller, and free-piston engine model. The rectifier controller provides PWM signals to the four power switch devices of the rectifier model. The PWM signals are calculated based on the current feedback of the motor, and a discrete proportional integral differential (PID) algorithm is used in this paper. The free-piston engine model is created according to the difference equations described in [16]. The free-piston engine model is used to output the piston motion trajectory to drive the linear motor finite element model. The model of the linear motor is shown in Figure 5. This model is a two-dimensional transient finite element model with a motion region. At this point, the mover is in the middle of the stroke, and the position is defined as the original point of the mover, as shown in Figure 5. The magnetic field is mainly distributed in ferromagnetic materials, and the leakage flux is relatively small. In order to display the internal structure, the meshes in the air box of the calculation region are hidden.

Combined Simulation Model
In order to analyze the electromagnetic loss of the designed PWM rectifier, its physical model is established by using the power electronics toolbox in the software of MATLAB. Supported by the finite element software we used to create the model of the linear motor, the finite element model of the linear motor is embedded into the MATLAB software. The MATLAB software calls the finite element software in every time steps, so the real combined simulation can be realized.
As shown in Figure 6, the combined model consists of four parts: linear motor finite element model, rectifier model, rectifier controller, and free-piston engine model. The rectifier controller provides PWM signals to the four power switch devices of the rectifier model. The PWM signals are calculated based on the current feedback of the motor, and a discrete proportional integral differential (PID) algorithm is used in this paper. The free-piston engine model is created according to the difference equations described in [16]. The free-piston engine model is used to output the piston motion trajectory to drive the linear motor finite element model.

Model Validation
A prototype of the linear motor system designed in the last section is developed. Figure 7 shows the prototype and its test platform. The test platform consists of five parts: the developed linear motor, prototype, the developed PWM rectifier, an air spring, a free piston engine and a battery.
A prototype of the linear motor system designed in the last section is developed. Figure 7 shows the prototype and its test platform. The test platform consists of five parts: the developed linear motor, prototype, the developed PWM rectifier, an air spring, a free piston engine and a battery.
The piston bore of the engine is 48 mm with the max stroke of 38 mm and the effective compression ratio of 7.8. It uses gasoline direct injection and spark ignition combustion under the excess air ratio of 1.1. The indicated power of the engine is about 1.5 kW, and the indicated thermal efficiency is 37% [17]. The piston bore of the air spring is 65 mm, which is bigger than the piston bore of the engine. The moving mass of both the piston and cylinder is about 0.5 kg.
A displacement sensor is connected to the mover of the linear motor, and a current sensor is installed in the alternating current side of the rectifier, so the real-time piston-mover position and the coil current of the linear motor can be tested. Based on the test platform, the combined model created in the last section is validated by comparing the simulated results with the prototype tested results. Figure 7. Prototype of the linear motor system and its test platform. Figure 8 shows the simulated and tested movement of the piston-mover assembly. As shown in the figure, a simplified testing cycle is used. The stroke of the testing cycle is 14 mm, and the reciprocating frequency is about 23 Hz. The testing cycle has two processes. In the first process, the linear motor works in motoring mode, the PWM rectifier works as an PWM inverter to drive the linear motor move from top to bottom, and the gas spring brakes the motion. In the second process, the gas spring drives the linear motor move from bottom to top, the PWM rectifier works in rectify and boost mode, and the linear motor works in generating mode to brake the motion.  The piston bore of the engine is 48 mm with the max stroke of 38 mm and the effective compression ratio of 7.8. It uses gasoline direct injection and spark ignition combustion under the excess air ratio of 1.1. The indicated power of the engine is about 1.5 kW, and the indicated thermal efficiency is 37% [17]. The piston bore of the air spring is 65 mm, which is bigger than the piston bore of the engine. The moving mass of both the piston and cylinder is about 0.5 kg.
A displacement sensor is connected to the mover of the linear motor, and a current sensor is installed in the alternating current side of the rectifier, so the real-time piston-mover position and the coil current of the linear motor can be tested. Based on the test platform, the combined model created in the last section is validated by comparing the simulated results with the prototype tested results. Figure 8 shows the simulated and tested movement of the piston-mover assembly. As shown in the figure, a simplified testing cycle is used. The stroke of the testing cycle is 14 mm, and the reciprocating frequency is about 23 Hz. The testing cycle has two processes. In the first process, the linear motor works in motoring mode, the PWM rectifier works as an PWM inverter to drive the linear motor move from top to bottom, and the gas spring brakes the motion. In the second process, the gas spring drives the linear motor move from bottom to top, the PWM rectifier works in rectify and boost mode, and the linear motor works in generating mode to brake the motion.

Model Validation
A prototype of the linear motor system designed in the last section is developed. Figure 7 shows the prototype and its test platform. The test platform consists of five parts: the developed linear motor, prototype, the developed PWM rectifier, an air spring, a free piston engine and a battery.
The piston bore of the engine is 48 mm with the max stroke of 38 mm and the effective compression ratio of 7.8. It uses gasoline direct injection and spark ignition combustion under the excess air ratio of 1.1. The indicated power of the engine is about 1.5 kW, and the indicated thermal efficiency is 37% [17]. The piston bore of the air spring is 65 mm, which is bigger than the piston bore of the engine. The moving mass of both the piston and cylinder is about 0.5 kg.
A displacement sensor is connected to the mover of the linear motor, and a current sensor is installed in the alternating current side of the rectifier, so the real-time piston-mover position and the coil current of the linear motor can be tested. Based on the test platform, the combined model created in the last section is validated by comparing the simulated results with the prototype tested results.  Figure 8 shows the simulated and tested movement of the piston-mover assembly. As shown in the figure, a simplified testing cycle is used. The stroke of the testing cycle is 14 mm, and the reciprocating frequency is about 23 Hz. The testing cycle has two processes. In the first process, the linear motor works in motoring mode, the PWM rectifier works as an PWM inverter to drive the linear motor move from top to bottom, and the gas spring brakes the motion. In the second process, the gas spring drives the linear motor move from bottom to top, the PWM rectifier works in rectify and boost mode, and the linear motor works in generating mode to brake the motion.    Figure 9 shows the simulated and tested coil current of the linear motor during the testing cycle. In each stroke, the coil current is controlled to standard sine wave by the PWM rectifier. The root mean square (RMS) currents in the two processes are 16.8 A and 9.8 A. The difference between the two RMS currents is due to the losses during the reciprocating motion cycle, including the electromagnetic loss discussed in the paper and the mechanical loss.
Electronics 2020, 9, x FOR PEER REVIEW 8 of 11 Figure 9 shows the simulated and tested coil current of the linear motor during the testing cycle. In each stroke, the coil current is controlled to standard sine wave by the PWM rectifier. The root mean square (RMS) currents in the two processes are 16.8 A and 9.8 A. The difference between the two RMS currents is due to the losses during the reciprocating motion cycle, including the electromagnetic loss discussed in the paper and the mechanical loss.  Figure 10 shows the simulated and tested PWM ratio of the PWM rectifier during the testing cycle. As shown in the figure, the results are consistent, especially when the linear motor system is operating in the power generation state. This consistency validates the combined model created in the last section, and indicates that the most electromagnetic loss of the linear motor system has been modeled by the combined model, so the combined model has enough precision to carry out the electromagnetic loss analysis in the next section.

Electromagnetic Loss Analysis
Based on the validated model in the last section, the electromagnetic loss and power conversion efficiency of the linear motor system designed for the FPEG are simulated and analyzed in this section. Considering that the linear motor system works in generating mode all the time during the stable running of the FPEG, a standard cycle is designed firstly according to the rated operating conditions of the FPEG. As shown in Figure 11, the linear motor system works in generating mode all the time during the standard cycle. The working frequency of the standard cycle is 50 Hz, which  Figure 10 shows the simulated and tested PWM ratio of the PWM rectifier during the testing cycle. As shown in the figure, the results are consistent, especially when the linear motor system is operating in the power generation state. This consistency validates the combined model created in the last section, and indicates that the most electromagnetic loss of the linear motor system has been modeled by the combined model, so the combined model has enough precision to carry out the electromagnetic loss analysis in the next section.
Electronics 2020, 9, x FOR PEER REVIEW 8 of 11 Figure 9 shows the simulated and tested coil current of the linear motor during the testing cycle. In each stroke, the coil current is controlled to standard sine wave by the PWM rectifier. The root mean square (RMS) currents in the two processes are 16.8 A and 9.8 A. The difference between the two RMS currents is due to the losses during the reciprocating motion cycle, including the electromagnetic loss discussed in the paper and the mechanical loss.  Figure 10 shows the simulated and tested PWM ratio of the PWM rectifier during the testing cycle. As shown in the figure, the results are consistent, especially when the linear motor system is operating in the power generation state. This consistency validates the combined model created in the last section, and indicates that the most electromagnetic loss of the linear motor system has been modeled by the combined model, so the combined model has enough precision to carry out the electromagnetic loss analysis in the next section.

Electromagnetic Loss Analysis
Based on the validated model in the last section, the electromagnetic loss and power conversion efficiency of the linear motor system designed for the FPEG are simulated and analyzed in this section. Considering that the linear motor system works in generating mode all the time during the stable running of the FPEG, a standard cycle is designed firstly according to the rated operating conditions of the FPEG. As shown in Figure 11, the linear motor system works in generating mode all the time during the standard cycle. The working frequency of the standard cycle is 50 Hz, which

Electromagnetic Loss Analysis
Based on the validated model in the last section, the electromagnetic loss and power conversion efficiency of the linear motor system designed for the FPEG are simulated and analyzed in this section. Considering that the linear motor system works in generating mode all the time during the stable running of the FPEG, a standard cycle is designed firstly according to the rated operating conditions of the FPEG. As shown in Figure 11, the linear motor system works in generating mode all the time during the standard cycle. The working frequency of the standard cycle is 50 Hz, which is equal to is equal to 3000 turns per minute of conventional engines. The coil current amplitude of the linear motor is 22 A, and the reciprocating stroke of the piston-mover movement is 36 mm.  Figure 12 shows the simulated instantaneous electromagnetic loss power in the linear motor of the standard cycle. Between the losses, the copper loss of coil accounts for 87% of the total loss. The copper loss depends on the winding wire diameter and is limited by the maximum allowable mass of the moving-coil, thus it is difficult to reduce. The eddy loss of the magnet is small, and this helps prevent the magnet from high temperatures and demagnetization. The eddy loss of the iron core is acceptable. This indicates that the electrical pure iron "DT4" is a suitable choice for the core material and sheets are not necessary. The figure also indicates that the aluminum alloy is not a good choice for the shell material, as the eddy loss is higher than expected.  Figure 13 shows the simulated instantaneous electromagnetic loss in the rectifier of the standard cycle. It can be seen that the loss power of each MOSFET is different, and it is related to their working states. Take the first half of the cycle to analyze. The EMF and current of the linear motor are positive, so the K3 is controlled by PWM signals and the others remain off. The current flows past the diode of K1 all the time, and its loss is the biggest. When the K3 PWM is off, the current flows past the diode of the K4, and its loss is the second biggest. No current flows past the K2 during the first half of the cycle, so its loss is zero. By analyzing the results, we can conclude that the diodes in the rectifier should be improved, as they account for most of the loss in the PWM rectifier.  Figure 12 shows the simulated instantaneous electromagnetic loss power in the linear motor of the standard cycle. Between the losses, the copper loss of coil accounts for 87% of the total loss. The copper loss depends on the winding wire diameter and is limited by the maximum allowable mass of the moving-coil, thus it is difficult to reduce. The eddy loss of the magnet is small, and this helps prevent the magnet from high temperatures and demagnetization. The eddy loss of the iron core is acceptable. This indicates that the electrical pure iron "DT4" is a suitable choice for the core material and sheets are not necessary. The figure also indicates that the aluminum alloy is not a good choice for the shell material, as the eddy loss is higher than expected.
Electronics 2020, 9, x FOR PEER REVIEW 9 of 11 is equal to 3000 turns per minute of conventional engines. The coil current amplitude of the linear motor is 22 A, and the reciprocating stroke of the piston-mover movement is 36 mm. Figure 11. Standard cycle for electromagnetic loss analysis. Figure 12 shows the simulated instantaneous electromagnetic loss power in the linear motor of the standard cycle. Between the losses, the copper loss of coil accounts for 87% of the total loss. The copper loss depends on the winding wire diameter and is limited by the maximum allowable mass of the moving-coil, thus it is difficult to reduce. The eddy loss of the magnet is small, and this helps prevent the magnet from high temperatures and demagnetization. The eddy loss of the iron core is acceptable. This indicates that the electrical pure iron "DT4" is a suitable choice for the core material and sheets are not necessary. The figure also indicates that the aluminum alloy is not a good choice for the shell material, as the eddy loss is higher than expected.  Figure 13 shows the simulated instantaneous electromagnetic loss in the rectifier of the standard cycle. It can be seen that the loss power of each MOSFET is different, and it is related to their working states. Take the first half of the cycle to analyze. The EMF and current of the linear motor are positive, so the K3 is controlled by PWM signals and the others remain off. The current flows past the diode of K1 all the time, and its loss is the biggest. When the K3 PWM is off, the current flows past the diode of the K4, and its loss is the second biggest. No current flows past the K2 during the first half of the cycle, so its loss is zero. By analyzing the results, we can conclude that the diodes in the rectifier should be improved, as they account for most of the loss in the PWM rectifier.  Figure 13 shows the simulated instantaneous electromagnetic loss in the rectifier of the standard cycle. It can be seen that the loss power of each MOSFET is different, and it is related to their working states. Take the first half of the cycle to analyze. The EMF and current of the linear motor are positive, so the K3 is controlled by PWM signals and the others remain off. The current flows past the diode of K1 all the time, and its loss is the biggest. When the K3 PWM is off, the current flows past the diode of the K4, and its loss is the second biggest. No current flows past the K2 during the first half of the cycle, so its loss is zero. By analyzing the results, we can conclude that the diodes in the rectifier should be improved, as they account for most of the loss in the PWM rectifier. Electronics 2020, 9, x FOR PEER REVIEW 10 of 11 Figure 13. Electromagnetic loss in the PWM rectifier. Table 3 lists the main parameters and the efficiency of the linear motor system designed by this paper for a FPEG. Under the standard working cycle of rated frequency 50 Hz and rated stroke 36 mm, the generating efficiency of the linear motor and the power conversion efficiency of the rectifier are 91.6% and 94.1%, respectively. The efficiency of the linear motor system is 86.3%. Considering the indicated thermal efficiency of engine is about 37% [17], the efficiency of the chemical-to-electrical energy conversion of about 31% can be obtained by the FPEG system, when it uses the motor system.

Conclusions
In this paper, a linear motor system composed of a moving-coil linear motor with axial magnetized magnets and an H-bridge PWM rectifier is designed for a portable free-piston engine generator. A combined electromagnetic simulation model is presented and validated by the prototype tested results. The electromagnetic loss of the linear motor system is analyzed by using the validated model. Under the rated working frequency of 50 Hz, the efficiency of 86.3% can be obtained by the proposed linear motor system, which meets the requirement of the design.
The electromagnetic analysis indicates that the copper loss is the main loss of the linear motor, and the diode loss is the main loss of the rectifier. The copper loss and the diode loss account for more than 90% of the total loss. In the future, we will try to improve the performance of the linear motor system by optimizing the moving-coil of the linear motor and using independent diodes to replace the internal diode of the MOSFET. We will test the efficiency of the linear motor system under the 50 Hz rated working cycle after manufacturing a more powerful mechanical drive system.  Table 3 lists the main parameters and the efficiency of the linear motor system designed by this paper for a FPEG. Under the standard working cycle of rated frequency 50 Hz and rated stroke 36 mm, the generating efficiency of the linear motor and the power conversion efficiency of the rectifier are 91.6% and 94.1%, respectively. The efficiency of the linear motor system is 86.3%. Considering the indicated thermal efficiency of engine is about 37% [17], the efficiency of the chemical-to-electrical energy conversion of about 31% can be obtained by the FPEG system, when it uses the motor system.

Conclusions
In this paper, a linear motor system composed of a moving-coil linear motor with axial magnetized magnets and an H-bridge PWM rectifier is designed for a portable free-piston engine generator. A combined electromagnetic simulation model is presented and validated by the prototype tested results. The electromagnetic loss of the linear motor system is analyzed by using the validated model. Under the rated working frequency of 50 Hz, the efficiency of 86.3% can be obtained by the proposed linear motor system, which meets the requirement of the design.
The electromagnetic analysis indicates that the copper loss is the main loss of the linear motor, and the diode loss is the main loss of the rectifier. The copper loss and the diode loss account for more than 90% of the total loss. In the future, we will try to improve the performance of the linear motor system by optimizing the moving-coil of the linear motor and using independent diodes to replace the internal diode of the MOSFET. We will test the efficiency of the linear motor system under the 50 Hz rated working cycle after manufacturing a more powerful mechanical drive system.