## 3 Modelling of CRPDC
### Modelling of Capacitive Reservoir Circuit
#### 3.1.1 Charging Modelling of CRPDC  
As shown in Figure 3c, the equivalent charging model of the CRPDC consists of the static resistance (\(R_{0}\)) of the P-MOSFET (Q1), the power-limiting resistance (\(R_{6}\)), the equivalent resistance and inductance of proportional valve coil (\(R_{4}\), \(L_{1}\)), the energy-storage capacitance (\(C_{1}\)), the current acquisition resistance (\(R_{5}\)), and the power voltage (\(V_{CC}\)). Based  
Figure 3: Charging and discharging work modes of the CRPDC: (**a**) Actual charging circuit; (**b**) actual discharging circuit; (**c**) charging equivalent model; (**d**) discharging equivalent model.  
on Kirchhoff's Current Law (KCL), the current relationship of the equivalent charging circuit can be written as:  
\[i=i_{\text{C}}+i_{L} \tag{1}\]  
where \(i\) denotes the main current, \(i_{\text{C}}\) denotes the capacitance current, and \(i_{L}\) denotes the inductance current.  
The main current \(i\) and capacitance current \(i_{\text{C}}\) can be obtained:  
\[i=\frac{V_{\text{CC}}-L_{1}\frac{di_{L}}{dt}-i_{L}(R_{4}+R_{5})}{R_{0}+R_{6}} \tag{2}\]  
\[i_{\text{C}}=C_{1}L_{1}\frac{d^{2}i_{L}}{dt^{2}}+C_{1}(R_{4}+R_{5})\frac{di_{L }}{dt} \tag{3}\]  
By substituting the main current Equation (2) and capacitance current Equation (3) into Equation (1), the differential equation for inductance current \(i_{L}\) can be formulated as:  
\[C_{1}L_{1}\frac{d^{2}i_{L}}{dt^{2}}+\bigg{[}\frac{L_{1}}{R_{0}+R_{6}}+C_{1}(R_ {4}+R_{5})\bigg{]}\frac{di_{L}}{dt}+\bigg{(}\frac{R_{4}+R_{5}}{R_{0}+R_{6}}+1 \bigg{)}i_{L}=\frac{V_{\text{CC}}}{R_{0}+R_{6}} \tag{4}\]  
With the differential Equation (4), the inductance current \(i_{L}\) can be derived as:  
\[i_{L}=K_{1}e^{\alpha_{1}t}+K_{2}e^{\alpha_{2}t}+\frac{V_{\text{CC}}}{R_{0}+R_ {4}+R_{5}+R_{6}} \tag{5}\]  
where  
\[\alpha_{1}=-\frac{1}{2C_{1}(R_{0}+R_{6})}-\frac{R_{4}+R_{5}}{2L_{1}}+\sqrt{ \bigg{[}\frac{1}{2C_{1}(R_{0}+R_{6})}+\frac{R_{4}+R_{5}}{2L_{1}}\bigg{]}^{2}- \frac{R_{0}+R_{4}+R_{5}+R_{6}}{(R_{0}+R_{6})C_{1}L_{1}}} \tag{6}\]  
\[\alpha_{2}=-\frac{1}{2C_{1}(R_{0}+R_{6})}-\frac{R_{4}+R_{5}}{2L_{1}}-\sqrt{ \bigg{[}\frac{1}{2C_{1}(R_{0}+R_{6})}+\frac{R_{4}+R_{5}}{2L_{1}}\bigg{]}^{2}- \frac{R_{0}+R_{4}+R_{5}+R_{6}}{(R_{0}+R_{6})C_{1}L_{1}}} \tag{7}\]  
The capacitance voltage \(u_{C}\) can be expressed as:  
\[u_{C}=L_{1}\frac{di_{L}}{dt}+i_{L}(R_{4}+R_{5}) \tag{8}\]  
By substituting the inductance current Equation (5) into the capacitance voltage Equation (8), the capacitance voltage \(u_{C}\) can be rewritten as:  
\[u_{C}=K_{1}(L_{1}\alpha_{1}+R_{4}+R_{5})e^{\alpha_{1}t}+K_{2}(L_{1}\alpha_{2 }+R_{4}+R_{5})e^{\alpha_{2}t}+\frac{V_{\text{CC}}(R_{4}+R_{5})}{R_{0}+R_{4}+R_ {5}+R_{6}} \tag{9}\]  
While the initial current \(i_{L0}\) of inductance \(L_{1}\) and the initial voltage \(u_{\text{C0}}\) of capacitance \(C_{1}\) are substituted into the inductance current Equation (5) and capacitance voltage Equation (9), respectively, the coefficients \(K_{1}\) and \(K_{2}\) can be derived as:  
\[K_{1}=\frac{u_{\text{C0}}-(R_{4}+R_{5})i_{L0}-L_{1}\alpha_{2}i_{L0}}{L_{1}( \alpha_{1}-\alpha_{2})}+\frac{\alpha_{2}V_{\text{CC}}}{(\alpha_{1}-\alpha_{2}) (R_{0}+R_{4}+R_{5}+R_{6})} \tag{10}\]  
\[K_{2}=\frac{u_{\text{C0}}-(R_{4}+R_{5})i_{L0}-L_{1}\alpha_{1}i_{L0}}{L_{1}( \alpha_{2}-\alpha_{1})}+\frac{\alpha_{1}V_{\text{CC}}}{(\alpha_{2}-\alpha_{1}) (R_{0}+R_{4}+R_{5}+R_{6})} \tag{11}\]  
Therefore, while the CRPDC is switched to be in charging circuit mode, the coefficients \(K_{1}\) and \(K_{2}\) can be determined by Equations (10) and (11). The following statesof inductance current \(i_{L}\) (coil current) and capacitance voltage \(u_{C}\) can be calculated by Equations (5) and (9), respectively.