## 2 Analysis of the FPLG System Working Process
### Linear Motor Modeling  
In this paper, the chosen linear motor is a cylindrical three--phase permanent magnet synchronous one. To precisely describe its power generation output characteristics, it's essential to conduct mathematical modeling and analysis. The voltage equation of the linear motor in the three--phase stationary ABC coordinate system is as follows:  
\[ \begin{cases}u_{a}=R_{s}\cdot i_{a}+L_{s}\cdot\frac{d_{ia}}{dt}+e_{a}\\ u_{b}=R_{s}\cdot i_{b}+L_{s}\cdot\frac{d_{ia}}{dt}+e_{b}\\ u_{c}=R_{s}\cdot i_{c}+L_{s}\cdot\frac{d_{ia}}{dt}+e_{c}\end{cases}  \tag{1}\]  
In the formula, \(u_{a}\), \(u_{b}\), and \(u_{c}\) are the phase voltages of the linear motor, \(R_{s}\) is the phase resistance of each phase winding, \(L_{s}\) is the phase inductance of each phase winding, and \(e_{a}\), \(e_{b}\), and \(e_{c}\) are the back electromotive forces. Their magnitude is directly correlated with the position of rotor, motion speed, and magnetic flux. Its mathematical expression is:  
\[\left\{ \begin{array}{l}e_{a}=-\psi_{f}\times\frac{\pi\times v}{\tau}\times sin (\theta+\theta_{0})\\ e_{b}=-\psi_{f}\times\frac{\pi\times v}{\tau}\times sin(\theta+\theta_{0}- \frac{2}{3}\pi)\\ e_{c}=-\psi_{f}\times\frac{\pi\times v}{\tau}\times sin(\theta+\theta_{0}+ \frac{2}{3}\pi)\end{array} \right. \tag{2}\]  
In the formula, \(\psi_{f}\) is the peak magnetic flux of the permanent magnet, \(v\) is the velocity of the rotor motion \(\theta\) From an electrical perspective \(\theta_{0}\) is the initial electrical angle. The core difference between linear and rotary motors lies in the mover's motion type. To compare them equivalently, ensure that within the same operating cycle, the linear motor's mover and the rotary motor's rotor travel the same displacement length. The conversion relationship is as follows:  
\[\left\{ \begin{array}{l}\frac{\theta\times r}{2p\pi\times r}=\frac{x}{2p\tau} \\ x=\frac{\theta\times\tau}{\pi}\end{array} \right. \tag{3}\]  
In the formula, \(p\) is the number of pole pairs of the linear motor \(\tau\) for polar distance \(\omega\) To induce the angle of electromotive force. The rotating motor scans \(p\) pairs of permanent magnets every time it rotates, which is equivalent to the displacement of \(p\) pairs of permanent magnet pole distances by the linear motor moving in one direction. Similarly, the conversion relationship between the speed of the linear motor rotor and the equivalent rotational speed is:  
\[v=\frac{\omega\times\tau}{\pi} \tag{4}\]  
Therefore, the general expression for the thrust of a linear motor is:  
\[F_{e}=\frac{3}{2}\times\frac{\pi}{\tau}\times\frac{\pi}{r}\times\left(\varphi _{d}i_{q}-\varphi_{q}i_{d}\right) \tag{5}\]  
Linear motors generally adopt a control strategy with rotor flux orientation \(i_{d}=0\), so that the electromagnetic thrust of the motor is only proportional to the amplitude of the stator current:  
\[F_{e}=\frac{3}{2}\times\frac{\pi}{\tau}\times\frac{\pi}{r}\times\varphi_{d}i_{ q}=K_{f}i_{q} \tag{6}\]  
In the formula, \(k_{f}\) is the electromagnetic thrust coefficient of the motor. The coupled dynamic model involved in the entire working process of the system exhibits nonlinear characteristics. During the motion of the piston mover assembly, various forms of energy flow through this assembly within the system, facilitating the transfer and conversion of energy. These energies collectively drive the piston mover assembly to perform cyclic motion. The structure and motion form of the linear motor are shown in Figure 2.  
Since the structure is not constrained by a crank-connecting rod mechanism, its motion is governed solely by the resultant force acting on the piston and mover. The piston mover assembly is subjected to the forces from the gas pressure in the two cylinders, frictional forces, and the thrust of the linear motor. The dynamic equation of the piston mover assembly is as follows:  
\[m\frac{d^{2}x}{d^{2}t}=F_{cl}+F_{cr}+F_{f}+F_{e} \tag{7}\]  
Figure 2: Linear motor mechanism and motion form.