## 2 Analytical Investigation of Electromagnetic Damper Model
### Magnetic Circuit and Operating Point
Equation (3) can be derived using the foregoing equation and assumptions:  
\[\oint_{\mathcal{C}}\overset{\rightarrow}{H}d\overset{\rightarrow}{l}=H_{m} \cdot l_{m}+2\int_{0}^{\tau_{m}}\frac{B_{p}}{\mu_{0}\mu_{Fe}}dr+2\int_{\tau_{ m}}^{\tau_{a}}\frac{B_{a}}{\mu_{0}}dr+2\int_{\tau_{a}}^{\tau_{c}}\frac{B_{c}}{ \mu_{0}\mu_{Cu}}dr+\frac{B_{s}}{\mu_{0}\mu_{Fe}}\big{(}l_{m}+l_{p}\big{)}\, \tag{3}\]  
where each subscript of \(B\) refers to the magnetic flux density in a region along the path. \(\mu_{Cu}\) and \(\mu_{Fe}\) indicate the relative permeabilities of copper and iron, respectively. The magnetic-field intensity can be expressed by the linearly approximated demagnetization line of the BH curve, according to Equation (4):  
\[H_{m}=\frac{B_{m}-B_{r}}{\mu_{m}},\ \ \mu_{m}=\frac{B_{r}}{H_{c}}\, \tag{4}\]  
where \(B_{m}\) represents the magnetic flux density of the permanent magnet, \(B_{r}\) represents the residual magnetic flux density, and \(H_{c}\) represents the coercive magnetic field. The BH curve is a magnetization curve that graphically describes the relationship between magnetic flux density and magnetic field strength. The Gaussian law of magnetic field can be expressed by Equation (5), under continuous conditions. \[\oint_{\mathcal{S}}\overset{\rightarrow}{B}\cdot ds=0. \tag{5}\]  
Equation (5) is employed to determine the magnetic flux density in different regions using the magnetic field of the magnet. The magnetic fields in each region (S1, S2, and S3) divided along the magnetic flux path in Figure 1 can be expressed as follows:  
\[B_{p}=\frac{B_{m}A_{m}}{A_{p}}=\frac{rB_{m}}{2l_{p}}\ (0\leq r\leq r_{m})\,\ \ B_{a}=\frac{B_{m}A_{m}}{A_{a}}=\frac{r_{m}^{2}B_{m}}{2 rl_{p}}\ (r_{m}\leq r\leq r_{a})\, \tag{6}\]  
\[B_{c}=\frac{B_{m}A_{m}}{A_{c}}=\frac{r_{m}^{2}B_{m}}{2 rl_{p}}\ (r_{a}\leq r\leq r _{c})\,B_{s}=\frac{B_{m}A_{m}}{A_{s}}=\frac{r_{m}^{2}B_{m}}{h_{y}(r_{c}+r_{s})}. \tag{7}\]  
As indicated by Equations (6) and (7), the magnetic fields in the air gap and coil are expressed in the same form. Therefore, the magnetic flux density equation can be derived by employing the same formula used for the radius range from the outside of the magnet to the coil. Equations (6) and (7) can be used to express each term in Equation (3) in terms of the magnetic flux density of the magnet:  
\[\int_{0}^{\tau_{m}}\frac{B_{p}}{\mu_{0}\mu_{Fe}}dr=\frac{r_{m}^{2}B_{m}}{4\mu_ {0}\mu_{Fe}l_{p}}=k_{p}B_{m}\,\ \ k_{p}=\frac{r_{m}^{2}B_{m}}{4\mu_{0}\mu_{Fe}l_{p}}\, \tag{8}\]  
\[\int_{\tau_{m}}^{\tau_{a}}\frac{B_{a}}{\mu_{0}}dr=\frac{r_{m}^{2}}{2\mu_{0}l_{ p}}\ln\Big{(}\frac{r_{a}}{r_{m}}\Big{)}B_{m}=k_{a}B_{m}\,\ \ k_{a}=\frac{r_{m}^{2}}{2\mu_{0}l_{p}}\ln\Big{(}\frac{r_{a}}{r_{m}}\Big{)}, \tag{9}\]  
Figure 1:0Figure 1:1  
By expressing the magnetic flux densities in the regions along the magnetic path in terms of the magnetic flux density of the magnet, the machine load line equation that determines the operating point of the permanent magnet can be derived. By substituting Equations (8)-(11) into Equation (3), Equation (12), i.e., the magnetic-field intensity equation for the permanent magnet and damper structure, is derived:  
Figure 1:2  
The machine load line equation and demagnetization curve determine the operating point. Using Equation (4), the approximate demagnetization curve is obtained, which gives the following flux density equation:  
Figure 1:3  
The magnetic flux density in Equation (13) is the amount of magnetic flux produced by the permanent magnet in the damper. The equation comprises the magnetic flux density, the permeability, and the main parameters of the damper. All the parameters defined in the damper model appear in the equation except for the air gap. Equations (12) and (13) define the operating point of the permanent magnet, which affects the behavior and performance of the electromagnetic damper.