## 3 Magnetic Negative Stiffness in QZS Isolators
### Mathematical Modeling of Permanent Magnet Negative Stiffness Elements
#### 3.2.3 Mechanical Modeling of Wedge-Shape Magnet Negative Stiffness Springs  
Figure 18 shows the three-dimensional and top view of the wedge-shaped permanent magnet, with the magnetization direction indicated as OO\({}_{1}\). The magnetic force between the surface unit \(dS_{i}\) and the surface unit \(dS_{j}\) can be expressed as follows:  
\[\mathrm{dF}=\frac{\mu_{0}}{4\pi}\frac{\sigma_{i}\sigma_{j}(\mathrm{P_{j}-P_{ i}})}{\left|\mathrm{P_{j}-P_{i}}\right|^{3}}\mathrm{dS_{i}}\mathrm{dS_{j}} \tag{28}\]  
where \(p_{i}\) and \(p_{j}\) are the position vectors of the face units \(dS_{i}\) and \(dS_{j}\), respectively, and \(\sigma_{i}\) is the electro-helicity density of the face i (i = 1, 2, 3, 4). Assuming that the rooted permanent magnet is an ideal model and is uniformly magnetized, the magnetic charge densities of the faces 5 and 6 are zero, whereas that of the other faces can be expressed as:  
\[\left\{ \begin{array}{l}\sigma_{1}=-\frac{1}{\mu_{0}}(\cos\alpha cos\beta+ \sin\alpha sin\beta)\\ \sigma_{2}=\frac{1}{\mu_{0}}(\cos\alpha cos\beta+\sin\alpha sin\beta)\\ \sigma_{3}=\sigma_{4}=\frac{1}{\mu_{0}}\sin\theta\\ \end{array} \right. \tag{29}\]  
where \(\beta\) is the angle between the y-axis and \(p_{i}\), \(J=B_{r}\). Thus the component of \(\mathrm{dF}\) in the \(Z\) direction can be expressed as:  
\[\mathrm{dF}_{z}=\frac{\mu_{0}}{4\pi}\frac{\sigma_{i}\sigma_{j}(\mathrm{h_{1}-h_{ 2}})}{\left|\mathrm{P_{j}-P_{i}}\right|^{3}}\mathrm{dS_{j}}\mathrm{dS_{j}} \tag{30}\]  
where \(h_{1}\) and \(h_{2}\) are the components of \(p_{i}\) and \(p_{j}\) in the \(Z\) direction, respectively. Figure 19 shows a pair of wedge-shaped permanent magnets \(PM^{M}\) and \(PM^{b}\) magnetized in the radial direction, and the magnetic force between the \(PM^{M}\) and \(PM^{b}\) surface cells can be obtained according to Equation (28). The magnetic force between face 1 of permanent magnet \(PM^{M}\) and face 1 of \(PM^{b}\) can be expressed as follows:  
\[ \begin{array}{l}\mathrm{F}_{M\mathrm{I}\mathrm{I}\mathrm{I}}^{ \mathrm{z}}=\frac{\mu_{0}}{4\pi}\int\frac{\sigma_{M\mathrm{I}}\sigma_{b\mathrm{ I}}(h_{1}-h_{2})}{|p_{b1}-p_{M\mathrm{I}}|^{3}}\mathrm{d}\mathrm{S}_{\mathrm{b} \mathrm{I}}\mathrm{d}\mathrm{S}_{M\mathrm{I}}\\ =\frac{\mu_{0}}{4\pi}\int_{\mathrm{M-M}}^{\mathrm{H+L}_{\mathrm{M}}}\int_{-L_ {b}}^{L_{b}}\delta_{a_{M}-\delta_{M}}^{a_{M}+\delta_{M}}\delta_{a_{b}-\delta_{ b}}^{a_{B}+\delta_{b}}\frac{\alpha_{M\mathrm{I}}\alpha_{\mathrm{I}}(h_{1}-h_{2})}{|p_{b1 }-p_{M\mathrm{I}}|^{3}}(\mathrm{R}_{1}+\mathrm{T}_{1})(\mathrm{R}_{2}+\mathrm{ T}_{2})\mathrm{d}h_{1}\mathrm{d}h_{2}\mathrm{d}\beta_{1}\mathrm{d}\beta_{2}\\ \mathrm{where}\\ \left\{\begin{array}{l}\sigma_{M\mathrm{I}}=\frac{1}{\mu_{0}}(\cos\alpha_{ \mathrm{M}}\cos\beta_{\mathrm{M}}+\sin\alpha_{\mathrm{M}}\sin\beta_{\mathrm{M }})\\ \sigma_{b\mathrm{I}}=-\frac{1}{\mu_{0}}(\cos\alpha_{\mathrm{b}}\cos\beta_{ \mathrm{b}}+\sin\alpha_{\mathrm{b}}\sin\beta_{\mathrm{b}})\\ |\mathrm{P}_{\mathrm{b}\mathrm{I}}-\mathrm{P}_{M\mathrm{I}}|^{2}=\left(\mathrm{ R}_{1}+\mathrm{T}_{1}\right)^{2}+\left(\mathrm{R}_{2}+\mathrm{T}_{2}\right)^{2}-2 \left(\mathrm{R}_{1}+\mathrm{T}_{1}\right)(\mathrm{R}_{2}+\mathrm{T}_{2})\cos (\beta_{2}-\beta_{1})+\left(h_{1}-h_{2}\right)^{2}\end{array}\right.\end{array}  \tag{32}\]  
Figure 19: Mathematical modeling of a pair of concentric wedge magnets: (**a**) three-dimensional view of a pair of concentric wedge magnets; and (**b**) top view of a pair of concentric wedge magnets. Figure 18: Mathematical modeling of single-wedge magnets: (**a**) three-dimensional view of a wedge magnet; and (**b**) top view of a wedge magnet. Deriving \(F_{M1b1}^{z}\) with respect to \(H\), the magnetic stiffness is given by  
\[K_{M1b1}(H)=\frac{dF_{M1b1}^{z}}{dH}\] \[=-\frac{h_{0}}{Z\tau}(\int_{-L_{b}}^{L_{b}}\int_{-L_{b}}^{\alpha_{M }+\phi_{M}}\int_{\alpha_{B}-\phi_{B}}^{\alpha_{B}+\phi_{B}}\frac{\sigma_{M1 \psi_{B1}}(h_{1}-h_{2})}{\left|p_{\mathrm{M1}}-p_{\mathrm{M1}}\right|^{3}}(R_{ 1}+T_{1})(R_{2}+T_{2})d\mathit{h}_{2}d\beta_{1}d\beta_{2})\left|