# One-Coil Long-Stroke Permanent Magnetic Actuator Design Applied to Load Breaker Switch for Railway
## 2 Permanent Magnet Actuator for Long-Stroke
### Design of the One-Coil Long Stroke PMA  
#### Analysis of the Magneto-Static Field and Prototype Design  
The circuit, motion, and magnetic field equations are required for the FEM analysis of an actuator. Figure 5 shows the equivalent circuit, the actuator's voltage source applies the electric current to a coil through electrostatic discharge by the capacitor, and Equation (1) is the equivalent circuit equation. Here, \(C\) is capacitance, \(R\) is resistance, and \(L\) is inductance.  
\[\frac{1}{C}\int i(t)=R\cdot i(t)+L\frac{i(t)}{dt} \tag{1}\]  
Equation (2) expresses the motion equation. Here, \(F\) is force of plunger, \(M\) is mass of driving parts, \(B\) is damping constant, \(K\) is spring constant, and \(F_{r}\) is frictional force.  
\[F=M\frac{d^{2}x}{dt^{2}}+B\frac{dx}{dt}+Kx+F_{r} \tag{2}\]  
Equations (3) and (4) express the magnetic field equation. Here, \(\overset{\rightarrow}{B}\) is magnetic flux density, \(\mu_{0}\) is permeability, \(\chi_{m}\) is magnetic susceptibility, \(\overset{\rightarrow}{H}\) is magnetic field intensity, \(\overset{\rightarrow}{J}\) is current density, and \(\overset{\rightarrow}{M}\) is magnetization.  
\[\overset{\rightarrow}{B}=\mu_{0}(1+\chi_{m})\overset{\rightarrow}{H}+ \overset{\rightarrow}{M} \tag{3}\]  
\[\frac{1}{\mu}\bigg{(}\
abla^{2}\overset{\rightarrow}{A}\bigg{)}=\overset{ \rightarrow}{J}+\frac{1}{\mu}\bigg{(}\
abla\times\overset{\rightarrow}{M} \bigg{)} \tag{4}\]  
Table 1 describes physical properties data applied to analysis [12; 13].  
Figure 4: The proposed PMA-type one-coil actuator.  
Figure 5: Equivalent circuit.  
Figure 6 shows the magnetic flux vector depending on the closing and opening operation of the one-coil actuator using the material properties described on Table 1. For the closing operation, the electromagnetic force from a coil compresses the open spring. This structure is efficient for the closing operation, as the direction of the magnetic flux vector by the permanent magnets is the same as that of a coil during closing. For the opening operation, electric current is applied to a coil in the opposite direction from the closing operation, offsetting the holding force with the magnetic flux vector by permanent magnets and that by a coil in the opposite direction. The opening operation is conducted with the force of the open spring compressing during the closing operation. The holding force can be offset by a small electric current, and the opening operation is mostly conducted by the force of the open spring. A single coil enables a compact design, and it can be operated by low electric power during the opening operation. As the spring is used for the opening operation, it has the disadvantage of requiring more energy to compress the spring for the closing operation.  
Figure 7 is a flow chart for one-coil long-stroke PMA development. The validity of the analysis was verified by making a prototype through FEM analysis and comparing the data. Through the test of the prototype, the parts that needed improvement were supplemented, and we confirmed whether the standards of the required specifications were satisfied.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Category** & **Material** & **Conductivity (S/m)** & **Permeability** & **Density (kg/m\({}^{3}\))** \\ \hline Core & SS41 & \(7\times 10^{6}\) & B-H curve & 7800 \\ Magnet & NdFe35 & 0.625\(\times\) 10\({}^{6}\) & 1.0652 & 7400 \\ Plunger & SS41 & \(7\times 10^{6}\) & B-H curve & 7800 \\ Coil & Copper & \(58\times 10^{6}\) & 0.999991 & 8960 \\ \hline \end{tabular}
\end{table}
Table 1: Material properties.  
Figure 6: Magnetic flux vector of one-coil actuator during (**a**) opening and (**b**) closing operations.