## 3 Magnetic Negative Stiffness in QZS Isolators
### Mathematical Modeling of Electromagnetic Negative Stiffness Elements
Consider a scenario where the energized coil has a length of \(L\), an inner diameter of \(R_{I}\), and an outer diameter of \(R_{2}\), as illustrated in \[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]1 Let the current passing through the coil be denoted as I, the number of turns per unit length as n, and the total number of turns in the thick-walled coil as \(N\). Initially, the analysis focuses on a single-layer coil with a radius of \(R\), as depicted in \[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]2 Since the energized helical coil exhibits symmetry, the analysis primarily concerns the magnetic field distribution in the yOz plane, where the y-axis represents the axial component. In the yOz plane, take any point \(Q(0,y_{0},z_{0})\), the y-axis coordinates y at the thickness of the ring, the current in the ring for \(I^{\prime}=nIdy\), in the ring on the \(S(R\cos\theta,y,R\sin\theta)\) point of the current element \(I^{\prime}d\overset{\rightarrow}{l}\). \[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]3  
The position vector \(\overset{\rightarrow}{r}\) from the current element \(I^{\prime}\mathrm{d}\overset{\rightarrow}{l}\) to Q is:  
\[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]4  
\[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]5 Schematic diagram of a spiral coil: (**a**) schematic diagram of the axial dimension combination; and (**b**) schematic diagram for combining radial dimensions. \[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]6 The position of a solenoid with an electric current. According to [PERSON]'s law, the magnetic induction \(\mathrm{d}\overrightarrow{B}\) produced by the current element \(I^{\prime}\mathrm{d}\overrightarrow{I}\) at point Q is:  
\[\mathrm{d}\overset{\rightarrow}{F}_{13}=\mathrm{d}\overset{\rightarrow}{F}_{1 3}\cdot\overset{\rightarrow}{e}_{y}=\frac{\sigma_{2}\sigma_{2}}{4\pi\mu_{0}} \frac{r_{1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}\mathrm{d}\mathrm{d} \mathrm{d}\beta}{\left|\overset{\rightarrow}{r}_{13}\right|^{3}}y_{0} \tag{37}\]7  
where \(r=\sqrt{R^{2}\cos^{2}\theta+\left(y_{0}-y\right)^{2}+\left(z_{0}-R\sin\theta \right)^{2}}\).