# Development of Proportional Pressure Control Valve for Hydraulic Braking Actuator of Automobile ABS
## 2 Analysis of Proportional Solenoid Force
### Solenoid Force of Proportional Electromagnet  
The total permeance of the proportional electromagnet is the reciprocal of the total reluctance, shown in Equation (7). It can be obtained by the series connection of air gaps and metal zones. When magnetomotive force exists, the magnetomotive force of air gaps can be obtained by the permeance and magnetic flux, and it varies with the armature position, as shown in Equation (8). Due to material properties, the magnetomotive force of metal zones is determined by the magnetic field intensity, as shown in Equation (9), although the permeability of steel (\(u_{Si}\)) cannot be obtained in advance. When the magnetic flux is obtained, the magnetic field intensity corresponds with the B-H curves. Then, the magnetomotive force can be calculated.  
\[P_{sum}=R_{sum}^{-1}=\left[P_{air}^{-1}+P_{steel}^{-1}\right]^{-1}=P_{air}P_{ steel}/(P_{air}+P_{steel}) \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(i\) & **Mean Path Area (\(A_{i}\))** & **Mean Path Length (\(l_{i}\))** & \(P_{Si}\) \\ \hline
1 & \(2\pi\)\(t_{s}R_{so}\) & \(L_{S2}+(L_{S1}+L_{S3})/2\) & \(u_{S}A_{1}/l_{1}\) \\
2 & \(\pi(R_{si}+R_{so}+t_{s}/2)\cdot L_{S1}\) & \(R_{so}+t_{s}/2\) & \(u_{S}A_{2}/l_{2}\) \\
3 & \(\pi(R_{B2}+R_{MO})\cdot L_{B1}\) & \(R_{B2}-(R_{M2}+R_{M0})/2\) & \(u_{S}A_{3}/l_{3}\) \\
4 & \(\pi(R_{B2}^{-}-R_{MO}^{2})\) & \(L_{B1}/2\) & \(u_{S}A_{4}/l_{4}\) \\
5 & \(\pi(R_{M2}^{-}-R_{MO}^{2})\) & \(L_{M}\) & \(u_{S}A_{5}/l_{5}\) \\
6 & \(\pi(R_{si}+R_{so}+t_{s}/2)\cdot L_{S3}\) & \(L_{S2}+(L_{S1}+L_{S3})/2\) & \(u_{S}A_{6}/l_{6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Permeance parameter of metal zone.  
Figure 9: Magnetic flux path of the metal zone.  
\[F_{air}=\Phi R_{air}=\Phi P_{air}{}^{-1} \tag{8}\]  
\[F_{steel}=\sum_{j=1}^{6}H_{j}l_{j}=\sum_{j=1}^{6}\frac{B_{j}}{u_{si}}l_{j}=\sum_{ j=1}^{6}\frac{\Phi}{u_{si}A_{j}}l_{j} \tag{9}\]  
\[NI=F_{sum}=F_{air}+F_{steel} \tag{10}\]  
In terms of the armature solenoid force, the input energy can be converted into magnetic energy and mechanical energy according to the conservation of energy, as shown in Equation (11). The current and the metal permeance are unrelated to armature displacement. When the stable current is input, the differential value of some items is zero. It can be derived from the principle of virtual work (\(W\)). The armature thrust can be obtained by the magnetic flux and permeance of gap zones and its derivation, as shown in Equation (12). A negative sign indicates the opposite direction of displacement force.  
\[W=\frac{1}{2}\Phi^{2}R_{air}=\frac{1}{2}\Phi^{2}P_{air}{}^{-1} \tag{11}\]  
\[\begin{array}{ll}F_{em}&=\frac{\partial W}{\partial x}=-\frac{\Phi^{2}}{2} \cdot\frac{d}{dx}(P_{air1}^{-1}+P_{air2}^{-1}+P_{air3}^{-1})\\ &=-\frac{\Phi^{2}}{2}\cdot[P_{air1}^{-2}\frac{d}{dx}P_{air1}+P_{air2}^{-2}\frac {d}{dx}P_{air2}+P_{air3}^{-2}\frac{d}{dx}P_{air3}]\end{array} \tag{12}\]  
As mentioned above, the armature solenoid force can be obtained by the magnetic flux and air gap parameter of Equations (2)-(4). The magnetic flux is an unknown input parameter since it has a nonlinear relationship with the input current, and it is obtained by the calculation program. The database of magnetic flux and magnetomotive force should be created, the actual magnetic flux is obtained through the input current. Then, the proportional solenoid force, as derived from Equation (11), is entered into Equation (12).