## 3 Magnetic Negative Stiffness in QZS Isolators
### Mathematical Modeling of Electromagnetic Negative Stiffness Elements
The projection of the micrometric magnetic force \(\mathrm{d}\overset{\rightarrow}{F}_{13}\) on the y-axis is:  
\[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]  
Ditto:  
\[\mathrm{d}\overset{\rightarrow}{F}_{14y}=-\mathrm{d}\overset{\rightarrow}{F}_ {14}\cdot\overset{\rightarrow}{e}_{y}=-\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{14}\right|^{3}}(y_{0}+l_{2}) \tag{38}\]  
\[\mathrm{d}\overset{\rightarrow}{F}_{23y}=-\mathrm{d}\overset{\rightarrow}{F}_ {23}\cdot\overset{\rightarrow}{e}_{y}=-\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{23}\right|^{3}}(y_{0}+l_{1}) \tag{39}\]  
\[\mathrm{d}\overset{\rightarrow}{F}_{24y}=\mathrm{d}\overset{\rightarrow}{F}_ {24}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{24}\right|^{3}}(y_{0}+l_{1}+ l_{2}) \tag{40}\]  
where  
\[\left|\overset{\rightarrow}{r}_{14}\right|=\sqrt{\left(y_{0}+l_{2} \right)^{2}+\left(r_{1}\sin\alpha-r_{2}\sin\beta\right)^{2}+\left(r_{1}\cos \alpha-r_{2}\cos\beta\right)^{2}}\] \[\left|\overset{\rightarrow}{r}_{23}\right|=\sqrt{\left(y_{0}+l_{1 }\right)^{2}+\left(r_{1}\sin\alpha-r_{2}\sin\beta\right)^{2}+\left(r_{1}\cos \alpha-r_{2}\cos\beta\right)^{2}}\] \[\left|\overset{\rightarrow}{r}_{24}\right|=\sqrt{\left(y_{0}+l_{1 }+l_{2}\right)^{2}+\left(r_{1}\sin\alpha-r_{2}\sin\beta\right)^{2}+\left(r_{1} \cos\alpha-r_{2}\cos\beta\right)^{2}}\]  
The total magnetic force along the y-axis is in a differential form:  
\[\mathrm{d}F_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}}\left(\frac{y_{0}}{ \left|\overset{\rightarrow}{r}_{13}\right|^{3}}+\frac{y_{0}+l_{1}+l_{2}}{ \left|\overset{\rightarrow}{r}_{24}\right|^{3}}-\frac{y_{0}+l_{2}}{\left| \overset{\rightarrow}{r}_{14}\right|^{3}}-\frac{y_{0}+l_{1}}{\left|\overset{ \rightarrow}{r}_{23}\right|^{3}}\right)r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2 }\mathrm{d}\mathrm{d}\beta \tag{41}\]  
Integrating the above equation yields:  
\[F_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}}\int\limits_{0}^{2\pi}\int\limits _{0}^{2\pi}\int\limits_{0}^{2\pi}\int\limits_{0}^{R_{2}}\left(\frac{y_{0}}{ \left|\overset{\rightarrow}{r}_{13}\right|^{3}}+\frac{y_{0}+l_{1}+l_{2}}{ \left|\overset{\rightarrow}{r}_{24}\right|^{3}}-\frac{y_{0}+l_{2}}{\left| \overset{\rightarrow}{r}_{14}\right|^{3}}-\frac{y_{0}+l_{1}}{\left|\overset{ \rightarrow}{r}_{23}\right|^{3}}\right)r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2 }\mathrm{d}\mathrm{d}\beta \tag{42}\]  
\[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]0 Relative positional relationship between electromagnets and permanent magnets: **(a)** section view of electromagnetic negative stiffness elements; and **(b)** side view of electromagnetic negative stiffness elements. Understanding the magnetic field distribution of a helical coil is pivotal for analyzing the electromagnetic force exerted on a permanent magnet within its magnetic field.