# Electromagnetic Characteristics Analysis of a Tubular Moving Magnet Linear Generator System
## 1 Introduction  
Free-piston engine (FFE) is a new type of electrical power generation system which is a combination of the traditional technology of internal combustion engines and the up-to-date technology of permanent-magnet linear generators. It is considered to have the advantages of simple structure, high efficiency, and good renewable fuel adaptability [1; 2; 3]. It can be used in multiple areas, such as electric vehicles, distributed power generation, and communication base stations. Figure 1 shows a typical composition of an opposed-piston FFE system. The system includes a central cylinder, two kickback air-springs, and two permanent-magnet linear generators. Compared with the traditional internal combustion engines with crankshafts, the pistons of FPE are without mechanical restraint, and they are connected to the mover of a linear generator. The simple structure makes the system more compact, lightweight, efficient, and reliable. The Sandia National Laboratory [4] has designed an opposed-piston FFE prototype. The prototype was tested to indicate that the synchronous movement of the two opposed pistons can be achieved by the control of the generating current. According to their testing results, the efficiency of the prototype can be comparable that of to fuel cells. A conversion efficiency of 50% from fuel to electricity can be obtained by the prototype when a relatively high compression ratio is used. The German Aerospace Center [5; 6] is currently studying the design of opposed-piston FPEs. The overall structure of the designed prototype is more compact than the prototype of Sandia National Laboratory. The performance of the prototype was tested using a test bench which used hydraulic pumps as the linear load. The researchers are trying to test the performance of a new designed prototype which uses two plate-moving magnet linear generators as the linear load. As the key component of FPEs, the performance of linear generators plays a very important role in the electric output and electric conversion efficiency of the system [7; 8]. The linear generator used in FPEs needs to have some strict and special characteristics, such as a good controllability, convenient manufacturing, low mover mass, and high generating efficiency [9]. These requirements make the design of linear generators for FPEs a great challenge [10; 11]. [PERSON] [12] designed a tubular three-phase moving magnet linear generator for FPEs. The electromagnetic characteristics of the linear generator were predicted by analytical expressions and finite element methods, but no prototype testing results were reported in the paper. This kind of three-phase moving magnet linear generator has the advantages of a higher efficiency and higher power density, but the generating process is difficult to control precisely. [PERSON] [13; 14] designed a tubular single-phase axial-magnetized moving magnet linear generator. The electromagnetic characteristics of the linear generator under a reciprocating frequency of 75 Hz were analyzed and tested. A generating efficiency of 82% was reported. This kind of single-phase moving magnet linear generator has advantages of being compact, lightweight, efficient, and reliable, but the generating efficiency should be improved to meet the requirement of FPEs. [PERSON] [15; 16; 17] designed a series of tubular single-phase moving coil linear generators for FPEs. This kind of single-phase moving coil linear generator has the advantage of having less moving mass, a faster response, and better controllability, but has the disadvantage of a lower power density. The copper loss is the main loss of this kind of linear generators and will lead to excessive temperatures of the moving coil. At the same time, the moving coil is difficult to cool. In addition to the above linear generator research, related linear actuator researches provided more references for this study. [PERSON] and [PERSON] [18; 19] designed a high-force short-stroke linear actuator for applications when a very high dynamic is required.