## 3 Characteristic Analysis of the Integrated Electric Hydraulic Vane Pump  
The proposed IEHVP comprises an outer rotor brushless DC motor and a balanced vane pump, as shown in Figure 5. The specific structure and operating principle are as follows:  
The proposed IEHVP combines the electric motor and the hydraulic pump, and its internal energy conversion form is shown in Figure 6. The electric energy enters the outer rotor motor and outputs mechanical energy through the internal magnetism chain, and then the balanced vane pump converts the mechanical energy into hydraulic energy. The output flow of the proposed IEHVP is determined by its outlet pressure and speed. When the externally applied voltage is constant, its output flow will change with the vibration of the outlet pressure; however, since the proposed IEHVP is a combination of an electric motor and a hydraulic pump, the operating characteristics of both need to be considered simultaneously. Ignoring the influence of temperature on motor performance and hydraulic oil performance, when the IEHVP is working, the loss of the internal motor output torque is as follows:  
\[\overline{T}_{av}=\left[ \begin{array}{cccc}1&1&1\end{array} \right]\left[ \begin{array}{c}T_{p}\\ T_{f}\end{array} \right]=\left[ \begin{array}{cccc}1&1&1&1&1\end{array} \right] \left[ \begin{array}{c}T_{r}\\ T_{b}\\ T_{th}\\ T_{v}\\ T_{t}\end{array} \right] \tag{1}\]  
where, \(T_{f}\) is the friction torque loss of the outer rotor motor; \(T_{r}\) is the air gap viscous friction torque of the outer rotor motor; \(T_{b}\) is torque loss caused by friction at the bearing; \(T_{p}\) is the input torque of the balanced vane pump; \(T_{th}\) is the theoretical torque of the balanced vane pump; \(T_{v}\) is the internal axial friction torque of the balanced vane pump; \(T_{t}\) is torque loss caused by friction between the top of the blade and the curved inner surface of the stator. As shown in Equation (1), when the proposed IEHVP is running, the output torque of the internal motor is mainly divided into two parts, one is the friction torque that needs to be overcome during operation, and the other is the required theoretical torque when the balanced blade pump is running. The methods for solving each torque are as follows:  
The air gap viscous friction torque of the outer rotor motor is:  
\[T_{r}=\frac{\mu\pi^{2}l_{0}{nd_{1}}^{3}}{120\delta_{2}} \tag{2}\]  
where, \(\delta_{2}\) is the thickness of the one-sided air gap, \(d_{1}\) is the motor rotor inner diameter, \(n\) is the motor rotor speed, and \(l_{0}\) is the axial length of air gap. Both ends of the outer rotor motor are respectively supported on the front and rear covers by needle roller bearings. The needle roller bearings only bear radial load, so the torque loss at the bearings is:  
\[T_{b}=\frac{f_{b}Gd_{2}}{2} \tag{3}\]  
where, \(f_{b}\) is the friction coefficient of the needle roller bearing; \(G\) is the equivalent load of the bearing; \(d_{2}\) is the nominal inner diameter of the bearing. Figure 6: The energy conversion flowchart of the proposed IEHVP. Theoretical torque of the balanced vane pump:  
\[T_{th}=\frac{pV}{2\pi} \tag{4}\]  
where, \(p\) is the outlet pressure of the integrated electric hydraulic pump. The flowing equation can obtain the internal axial friction torque of the balanced vane pump:  
\[T_{v}=2T_{1}=\frac{\mu n\pi^{2}}{60\delta_{1}}(r_{0}+r_{4})^{2}\Big{(}r_{4}^{2} -r_{0}^{2}\Big{)} \tag{5}\]  
According to the literature [23; 24; 25], during the operation of the balanced vane pump, the contact force between the blade tip and the inner surface of the stator changes periodically. Therefore, the average value of the contact force between a single blade and the stator surface within one revolution and the total torque generated by the friction between all blades and the stator are respectively:  
\[\left\{ \begin{array}{l}F_{t}=\frac{Sp_{0}}{2}\\ T_{t}=Z_{f}t_{t}F_{t}\frac{(r_{0}+r)}{2}\end{array} \right.