## 2 Preliminary Consideration
The above changes continually with the pump outlet pressure at the left end of the constant power valve spool. The displacement of the variable pump decreases with the increase in the pump outlet pressure. At this time, the balance formula of the spool can be deduced as  
\[P_{\rm p}S_{3}=P_{1}S_{3}+k_{1}\Delta x_{1}+(k_{1}+k_{2})\Delta x_{2} \tag{3}\]  
where \(k_{2}\) is the elastic coefficients of spring 2. \(\Delta x_{2}\) is the compression displacement of spring 2. Until the displacement of variable displacement pump is adjusted to the minimum displacement. Because the pump displacement changes with the pump outlet pressure, the constant power valve spool works in the process from one spring to two springs, and the spring elasticity coefficient changes, which affects the slope of the pump displacement changing with the pump outlet pressure. When the input speed of the pump is constant. The variation curve of the pump outlet flow with the pump outlet pressure is shown in Figure 2.  
As can be seen, the approximate constant power control is realized by changing the spring stiffness. The pump enters the constant power mode from point \(A\). The pressure at point \(A\) depends on the starting pressure set by the spring 1. Starting from point \(B\), the constant power enters the second stage, which is adjusted by the springs 1 and 2. The fitting curve of approximate constant power can be obtained by the joint action of two springs. After entering the constant power point of the pump, the displacement of the variable pump varies linearly with the displacement of the valve core of the constant power valve, which can be expressed as  
\[\Delta V=\lambda\Delta x \tag{4}\]  
where \(\Delta V\) is the variation of the displacement of the pump. \(\lambda\) is the scale factor. \(\Delta x\) is the displacement of the valve core of the constant pressure valve. When the pump operates in the section \(AB\), the flow of the pump can be given as  
\[Q=Q_{\rm max}-\Delta Q=(V_{\rm max}-\Delta V)n \tag{5}\]  
where \(Q\) is the output flow of the pump. \(Q_{\rm max}\) is the maximum flow of the pump. \(\Delta Q\) is the reduced flow of the pump due to the reduction in displacement of the pump. \(V_{\rm max}\) is the maximum displacement of the pump. \(\Delta V\) is the reduction in displacement of the pump. \(n\) is the speed of the pump. Figure 2: Outlet pressure variation with flow curve of constant power pump. Combined Equations (2) and (5), the relationship between outlet flow \(Q\) and pressure \(P_{\text{P}}\) of the pump in section AB can be deduced as  
\[Q=-\lambda nS_{3}\Big{/}_{k_{1}}P_{\text{P}}+(nV_{\text{max}}+\left.\lambda nP_{ 1}S_{3}\Big{/}_{k_{1}}\right) \tag{6}\]  
The constant value in Equation (6) is defined as  
\[\left\{ \begin{array}{l}K_{1}=\frac{\lambda nS_{3}}{k_{1}}\\ C_{1}=nV_{\text{max}}+\frac{\lambda nP_{1}S_{3}}{k_{1}}\end{array} \right. \tag{7}\]  
Combined Equations (6) and (7), the relationship between outlet flow \(Q\) and pressure \(P_{\text{P}}\) of pump in section AB can be re-expressed as  
\[Q=-K_{1}P_{\text{P}}+C_{1} \tag{8}\]  
Through Equation (6), when the hydraulic pump enters the \(AB\) stage of constant power operation of the pump from point A, the pump outlet flow \(Q\) has a negative linear relationship with the pressure \(P_{\text{P}}\), and the slope is \(-K_{1}\). In the section \(BC\), the displacement of the core valve of the constant pressure valve is \(\Delta\text{x}_{1}\). The further movement displacement of the core valve of the constant pressure valve is \(\Delta\text{x}_{2}\). Combine Equation (4), the outlet flow of the pump can be given as  
\[Q=-K_{1}P_{\text{P}}+C_{1} \tag{9}\]  
Combine Equations (3) and (5), the relationship between outlet flow \(Q\) and pressure \(P_{\text{P}}\) of pump in section \(AB\) can be deduced as  
\[Q=\frac{\lambda nS_{3}}{k_{1}+k_{2}}P_{\text{P}}+(nV_{\text{max}}+\frac{ \lambda n(P_{1}S_{3}+k\Delta x_{1})}{k_{1}+k_{2}}-\lambda n\Delta x_{1}) \tag{10}\]  
The constant value in Equation (10) is defined as  
\[\left\{