## 2 Materials and Methods: Architectural Configurations for the Case Study
### The Hydraulic System
#### 2.1.3 Theoretical Analysis of a Valve-Controlled Circuit
\[p_{A}^{o}=p_{P}-\Delta p_{n}\bigg{(}\frac{vA_{P}}{Q_{n}}\bigg{)}^{2} \tag{5}\]  
\[p_{B}^{o}=\Delta p_{n}\bigg{(}\frac{vA_{P}}{\alpha Q_{n}}\bigg{)}^{2} \tag{6}\]  
The thrust force created by the rod is expressed by Equation (7). \[F^{o}=p_{A}^{o}A_{P}-p_{B}^{o}\frac{A_{P}}{\alpha}=p_{P}A_{P}-A_{P}\Delta p_{n }\bigg{(}\frac{vA_{P}}{Q_{n}}\bigg{)}^{2}-\frac{A_{P}}{\alpha}\Delta p_{n} \bigg{(}\frac{vA_{P}}{\alpha Q_{n}}\bigg{)}^{2} \tag{7}\]  
For inward motion, a similar analysis procedure is used as for outward motion, with flow rates, pressure drops on the valve paths, and pressures in the cylinder chambers determined as in Equations (8)-(12). \[Q_{B}^{i}=vA_{T}=\frac{vA_{P}}{\alpha}\hskip 28.452756 ptQ_{A}^{i}=\alpha Q_{B} ^{i}=vA_{P} \tag{8}\]  
\[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]0  
\[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]1  
\[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]2  
\[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]3  
In the inward phase, the ratio between the pressure drops between ports P and B and A and T is \(\frac{\Delta p_{BT}}{\Delta p_{AT}}=\frac{1}{\alpha^{2}}=\frac{1}{4}\). During the retraction phase, the stem exerts a pulling action. This force is expressed by Equation (13). \[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]4  
With \(v\) and \(p_{P}\) being equal, the thrust force is greater than the pulling force. Both in the exit and retraction phases, the maximum force value is obtained for \(v=0\). In this case, the ratio between \(F^{o}\) and \(F^{i}\) is equal to \(\alpha\). \[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]5  
If \(Q_{P}\) is the effective flow rate delivered by the pumping group, the maximum translation speed of the rod during the output stroke is the minimum value between the one obtained from Equation (1) setting \(Q_{A}^{o}=Q_{P}\) and the one that derives from setting \(F^{o}\) equal to zero. \[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]6Similarly, the maximum translational velocity of the rod during the return stroke is the minimum value between the value obtained from Equation (8) when \(Q_{B}^{i}=Q_{P}\) is set and the value obtained when the force \(F^{i}\) is set equal to zero. \[Q_{A}^{o}=vA_{P}\hskip 28.452756 ptQ_{B}^{o}=\frac{Q_{A}^{o}}{\alpha} \tag{1}\]7  
Regeneration usually occurs during rod extension and consists in directing the oil flow, which exits the front chamber of the actuator as a result of the translational motion of the piston, into the sleeve of the rear chamber cylinder. This results in a sensitive energy saving, which is why architectures that use DCVs with regeneration are called economizer circuits. In the proposed study, the point where the fluid is re-supplied is downstream of the PA path, i.e., towards the user, therefore, the regeneration is of type A, and A--hybrid. The configuration with regeneration A--hybrid integrates in the valve body the functions of a check valve in the line BA and an on-off valve in the connection between point A and the tank. These two functions allow it to go from a configuration with internal regeneration--A to a standard configuration during the discharge stroke. In this way, it is possible to take advantage of both the benefits of internal regeneration--A (energy efficiency) and the benefits of the standard configuration (high forces), if necessary. The analysis of flow rates, pressures, velocities, and forces for the outward stroke in the case of regeneration A--internal follows Equations (17)-(21), in a similar manner as for the standard configuration.