## 2 Mathematical Model Analysis
### Mathematical Model of Positive Flow System  
The traditional positive flow system applied to a traditional CM uses a variable pump with a constant speed system [1]. Due to the excellent motor speed response, the quantitative pump-variable speed positive flow system was investigated. The principle of the quantitative pump-variable speed positive flow system is shown in Figure 1.  
Figure 1: Simplified principle diagram of quantitative pump-variable speed positive flow system for an excavator boom. The target speed of the electric motor pump is controlled by the pilot handle output signal. The greater the pilot output signal is, the greater the electric motor speed achieves. To prevent the system from overloading, relief valves are equipped in both chambers of the oil cylinder. A check globe valve is set up to avoid the cylinders falling because of the gravity when the multi-way valve is in the neutral position. The mathematic model of the hydraulic system is analyzed. To simplify the analysis, some assumptions are provided as:  
* Hydraulic pump inlet pressure, system return pressure, and the tank pressure are assumed to be zero;
* The relief valve is not set at the pump outlet port. The leakage of hydraulic pump, valves, and actuators is ignored;
* The flow fluctuation caused by pipeline pressure loss and pipeline friction is not considered. The output flow equation of the pump can be expressed as:  
\[q_{\mathrm{V_{1}}}=V_{1}n-C_{\mathrm{P}}p_{1} \tag{4}\]  
where \(q_{\mathrm{V_{1}}}\) is the pump output flowrate; \(V_{1}\) is the pump displacement; \(n\) is the pump speed; \(C_{\mathrm{P}}\) is the pump leakage factor; and \(p_{1}\) is the pump output pressure. The relationship between the output pressure and the torque of the pump can be deduced as:  
\[p_{1}=\frac{2\pi\cdot T}{V_{1}} \tag{5}\]  
where \(T\) is the pump output torque. The force balance equation of the globe valve when the cylinder extends is:  
\[p_{2}-p_{3}=k_{1}x_{1}/A_{1} \tag{6}\]  
where \(p_{2}\) is the output pressure of the check valve; \(p_{3}\) is the pressure of the cylinder in a rodless cavity; \(k_{1}\) is the elastic coefficient of the globe valve in the spring cavity; \(x_{1}\) is the spring displacement of the globe valve; and \(A_{1}\) is the effective area of the globe valve in the spring cavity. The force balance equation of the globe valve when the cylinder retracts can be provided as:  
\[p_{3}-p_{4}=k_{1}x_{1}/A_{1} \tag{7}\]  
where \(p_{4}\) is the inlet pressure of orifice in the multi-way valve. The flow equation of orifice in the multi-way valve can be written as:  
\[q_{\mathrm{V_{2}}}=C_{\mathrm{q}}W_{\mathrm{v}}X_{\mathrm{v}}\sqrt{\frac{2}{ \rho}\cdot\Delta p} \tag{8}\]  
where \(q_{\mathrm{V_{2}}}\) is the flowrate through the orifice; \(C_{\mathrm{q}}\) is the flow coefficient; \(W_{\mathrm{v}}\) is the flow area gradient; \(X_{\mathrm{v}}\) is the orifice opening size; \(\rho\) is the oil viscosity; and \(\Delta p\) is the pressure difference before and after orifice. The force balance equation of cylinder can be expressed as:  
\[p_{3}A_{2}=p_{4}A_{3}+m\frac{\mathrm{d}v}{\mathrm{d}t}+b_{\mathrm{L}}v+F \tag{9}\]  
where \(A_{2}\) is the effective area of the cylinder in a rodless cavity; \(p_{4}\) is the pressure of the cylinder in a rod cavity; \(A_{3}\) is the effective area of the cylinder in a rod cavity; \(m\) is the mass of the piston and the load converted to the piston; \(v\) is the cylinder speed; \(b_{\mathrm{L}}\) is the viscous damping of the piston and load; and \(F\) is the load force. It can be seen from the mathematical model of the positive flow system that the performance of the pump outlet flowrate is closely related to the electric motor speed performance which directly affects the speed control performance of the positive flow system. Furthermore, pump outlet pressure is closely related to the pump input torque which directly affects the pump output pressure characteristics.