## 2 Preliminary Consideration  
In the traditional excavator based on engine, the speed of engine is preset and fixed during operation. To match the operation requirements of different working conditions and improve the absorption rate of the hydraulic pump from the engine output power, the constant power control is employed to limit the hydraulic system output power and match the output power of power source. The constant power control is adjusted through the displacement regulation mechanism of pump. The structural principle diagram of the constant power regulation system is given in Figure 1.  
The constant power valve is a three-position and three-way directional valve. The valve sleeve is connected with the piston of the pump displacement variable mechanism. The left end of the constant pressure valve core is connected with the pump outlet pressure signal \(P_{\rm p}\). Two springs are equipped at the right end, with lengths of \(L_{1}\) and \(L_{2}\), respectively, where \(L_{1}>L_{2}\) and elastic coefficients are \(k_{1}\) and \(k_{2}\), respectively. The initial state of spring 1 is compressed. The initial set pressure of spring 1 is \(P_{1}\). The spring 2 is naturally extended without pressure. When \(P_{\rm p}<P_{1}\), the constant power valve spool moves to the left. The constant power valve works in the right position. The oil chamber at the left end of the variable piston returns to the oil tank. The variable piston moves to the left. The variable pump works at the maximum displacement. When \(P_{\rm p}=P_{1}\), the constant power valve works in the middle position. The oil chamber at the left end of the variable piston is closed. The piston displacement remains unchanged, and the variable pump still works at the maximum displacement. When \(P_{\rm p}>P_{1}\), the constant power valve core moves to the right. The constant power valve works in the left position. The oil chamber at the left end of the variable piston is connected to the pump outlet. Suppose the left end cross-sectional area of the variable piston is \(S_{1}\) and the right end cross-sectional area of the variable piston is \(S_{2}\), and \(S_{1}>S_{2}\). The forces on the left and right ends of the variable piston can be expressed as  
\[T_{1}=P_{\rm p}S_{1}>P_{\rm p}S_{2}=T_{2} \tag{1}\]  
where \(T_{1}\) is the force of the variable piston left end. \(T_{2}\) is the force of the variable piston right end. \(P_{\rm p}\) is the output pressure of the pump. \(S_{1}\) is the left end cross-sectional area. \(S_{2}\) is the right end cross-sectional area. The variable displacement piston moves to the right to reduce the displacement of the pump. Meanwhile, the variable piston moving to the right drives the constant power valve sleeve to the right through the connecting rod so that the constant power valve works in the middle position again. The variable pump maintains the current displacement. The compression displacement of spring 1 is \(\Delta x_{1}\), and the cross-sectional area of the left end of the valve core is \(S_{3}\). At this time, the balance formula of the constant power valve core can be given as  
\[P_{\rm p}S_{3}=P_{1}S_{3}+k_{1}\Delta x_{1} \tag{2}\]  
where \(S_{3}\) is the area of the left end of the constant power valve core. \(P_{1}\) is the initial set pressure of spring 1. \(k_{1}\) is the elastic coefficients of spring 1. \(\Delta x_{1}\) is the compression displacement of spring 1. When the load pressure continues to rise, causing the rise of pump outlet pressure \(P_{\rm p}\). The constant power valve continues to repeat the above steps. The displacement of the  
Figure 1: Structural principle diagram of constant power regulation system. variable pump decreases with the increase in the pump outlet pressure until the valve core of the constant power valve contacts the spring 2. The compression displacement of spring 2 is \(\Delta x_{2}\). At this time, spring 1 and spring 2 act together on the right side of the constant power valve spool. The elastic coefficient of these two springs is \(k_{1}+k_{2}\).