## 2 Analytical Investigation of Electromagnetic Damper Model
### Winding Properties and Damping Force  
The winding resistance affects the damping force of the damper together with the external variable resistance for semi-active control. When current flows in a closed-loop coil, the properties of the winding must be considered. Owing to the finite conductivity of the winding and [PERSON]'s law, a voltage drop occurs in the coil and energy is dissipated. Additionally, the terminal voltage of the damper is lower than the voltage induced by the movement of the driver. The axial flux generated in the coil by the induced current acts as a circuit inductance. The inductance causes a delay between the induced electromotive force and the current, resulting in a delay and phase difference between the external and reaction force.  
The resistance of the coil is calculated using the total number of turns in the coil and the average radius. Because all the coils are connected in series, the total winding resistance is calculated as follows:  
\[R_{c}=\frac{r_{a}+r_{c}}{r_{w}^{2}\rho}pN_{w}\, \tag{18}\]  
where \(r_{w}\) represents the radius of the coil section, \(N_{w}\) represents the total number of turns in the coil, \(\rho\) represents the conductivity of copper as the material of the coil, and \(p\) represents the number of coils or poles.  
Generally, the inductance changes with the magnetic flux, depending on the relative position of the driver inside the winding. However, if the length of the mover in the axial direction (in which the driver moves) is equal to the length of the coil and the stroke length is limited, avoiding the poles at the edge of the winding, the inductance can be approximated as follows:  
\[L_{c}=\frac{\mu_{0}\mu_{Fe}\pi r_{a}^{2}}{2l_{p}+2\mu_{Fe}\left(l_{c}-l_{p} \right)}pN_{w}^{2}. \tag{19}\]  
To quantify the force acting on the driver owing to the current inside the coil, the Lorentz force can be applied. The Lorentz force is calculated by integrating the product of the current density and the magnetic field with respect to the wire volume \(V\), as follows:  
\[\mathbf{F}=\int_{V}(\mathbf{J}\times\mathbf{B})d\upsilon. \tag{20}\]  
Equation (20) can be modified to express the force applied to a single wire:  
\[F_{w}=\int_{l}(JAB_{c})dl=\int_{l}iB_{c}dl=2\pi riB_{c}\, \tag{21}\]  
where \(l\) represents the length of the wire, \(A\) represents the cross-sectional area of the wire, \(B_{c}\) represents the magnetic flux density inside the coil, and \(i\) represents the current inside the coil. The current induces an axial magnetic flux in the coil. The induced magnetic flux is perpendicular to the magnetic flux of the air gap produced by the magnet. By substituting the left equation of Equation (7)into Equation (21), the force of a single wire can be expressed according to the magnetic flux density of the magnet:  
\[F_{w}=\frac{\pi r_{m}^{2}B_{m}}{l_{p}}i. \tag{22}\]  
Therefore, the total damping force is derived as follows:  
\[F_{d}=pN_{a}\frac{\pi r_{m}^{2}B_{m}}{l_{p}}i=k_{f}\cdot i\,\ \ k_{f}=pN_{a} \frac{\pi r_{m}^{2}B_{m}}{l_{p}}i\, \tag{23}\]  
where \(k_{f}\) is the force constant of the device. In the case of a passive or semi-active damper, the circuit current and hence the damping force are determined by the circuit connected to the coils of the damper and the voltage induced by the translational motion of the mover.