## 3 Magnetic Negative Stiffness in QZS Isolators
### Mathematical Modeling of Permanent Magnet Negative Stiffness Elements
#### 3.2.2 Mechanical Modeling of Rectangular Magnet Negative Stiffness Springs
They are aligned parallel to the 2c side and possess uniform magnetic densities denoted as \(\sigma\) and \(\sigma^{\prime}\) in their upper and lower planes, respectively. It is assumed that the dimensions are \((2a,2b,2c)\) and \((2a^{\prime},2b^{\prime},2c^{\prime})\) for M1 and M2, and that the magnetization strengths of each magnet are denoted as \(J\) and \(J^{\prime}\), respectively, with the two magnets magnetized in opposite directions. This magnetic charge density can be expressed as:  
\[\sigma=\overset{\rightarrow}{J}\cdot\overset{\rightarrow}{N} \tag{19}\]  
At this time, the upper and lower relative two planes \(2a\times 2b\) and \(2a^{\prime}\times 2b^{\prime}\), and the upper and lower planes have a uniform magnetic density \(\sigma\) and \(\sigma^{\prime}\), then the upper and lower two permanent magnets in the vertical direction of the static magnetic energy can be expressed as:  
\[E=\int_{-a}^{a}dx\overset{\rightarrow}{\int_{-b}^{b}dy\int_{-a^{\prime}}^{a^{ \prime}}dx^{\prime}\int_{-b^{\prime}}^{b^{\prime}}dy\frac{JJ^{\prime}}{4\pi \mu_{0}r}} \tag{20}\]  
where \(r=\sqrt{\left(\alpha+a^{\prime}-a\right)^{2}+\left(\beta+y^{\prime}-y\right)^ {2}+\gamma^{2}}\). The quadratic integration of the above equation can be expressed as:  
\[E=\frac{JJ^{\prime}}{4\pi\mu_{0}}\sum_{i=0}^{1}\sum_{j=0}^{1}\sum_{k=0}^{1} \sum_{l=0}^{1}\sum_{p=0}^{1}\sum_{q=0}^{1}\left(-1\right)^{i+j+k+l+p+q}\phi( \mu_{ij},\
u_{kl},\omega_{pq},r) \tag{21}\]  
where  
\[\phi(\mu,\
u,\omega,r)= \begin{array}{l}\frac{1}{2}\mu(\
u^{2}-\omega^{2}) \ln(r-\mu)+\frac{1}{2}\
u(\mu^{2}-\omega^{2})\ln(r-\
u)\\ +\mu\
u\omega\arctan\frac{\mu\
u}{r\omega}+\frac{r}{6}(\mu^{2}+v^{2}-2\omega^ {2})\end{array}  \tag{22}\]  
\[\left\{ \begin{array}{l}\mu_{ij}=\alpha+(-1)^{j}a^{\prime}-(-1)^{i}a\\ \
u_{kl}=\beta+(-1)^{l}b^{\prime}-(-1)^{k}b\\ \omega_{pq}=\gamma+(-1)^{q}c^{\prime}-(-1)^{p}c\\ r=\sqrt{\mu_{ij}^{2}+v_{kl}^{2}+\omega_{pq}^{2}}\end{array} \right. \tag{23}\]  
Figure 17: Mathematical modeling of negative stiffness in a magnet’s negative stiffness structures. According to the principle of the equivalent magnetic charge model, the interaction force F between the two permanent magnets can be obtained along the \(X\), \(Y\), and \(Z\) directions, respectively, as:  
\[\left\{ \begin{array}{l}F_{x}=\frac{JJ^{\prime}}{4\pi\eta_{0}}\sum\limits_{i=0}^{ 1}\sum\limits_{j=0}^{1}\sum\limits_{k=0}^{1}\sum\limits_{l=0}^{1}\sum\limits_{ p=0}^{1}\sum\limits_{q=0}^{1}\left[(-1)^{i+j+k+l+p+q}\varphi_{x}(\mu_{ij},\
u_{kl}, \omega_{pq},r)\right]\\ F_{y}=\frac{JJ^{\prime}}{4\pi\eta_{0}}\sum\limits_{i=0}^{1}\sum\limits_{j=0}^{ 1}\sum\limits_{k=0}^{1}\sum\limits_{k=0}^{1}\sum\limits_{p=0}^{1}\sum\limits_{ q=0}^{1}\left[(-1)^{i+j+k+l+p+q}\varphi_{y}(\mu_{ij},\
u_{kl},\omega_{pq},r) \right]\\ F_{z}=\frac{JJ^{\prime}}{4\pi\eta_{0}}\sum\limits_{i=0}^{1}\sum\limits_{j=0}^{ 1}\sum\limits_{k=0}^{1}\sum\limits_{k=0}^{1}\sum\limits_{l=0}^{1}\sum\limits_{ p=0}^{1}\sum\limits_{q=0}^{1}\left[(-1)^{i+j+k+l+p+q}\varphi_{z}(\mu_{ij},\
u_{kl}, \omega_{pq},r)\right]\\ \end{array} \right. \tag{24}\]  
where  
\[\left\{ \begin{array}{l}\varphi_{x}(\mu_{ij},\
u_{kl},\omega_{pq},r)=\frac{ \
u^{2}-\omega^{2}}{2}\ln(r-\mu)+\mu\
u\ln(r-\
u)+\
u\omega\text{arctan}\frac {\mu\
u}{r\omega}+\frac{\tau_{ll}}{2}\\ \varphi_{y}(\mu_{ij},\
u_{kl},\omega_{pq},r)=\frac{\
u^{2}-\omega^{2}}{2}\ln( r-\
u)+\mu\
u\ln(r-\mu)+\
u\omega\text{arctan}\frac{\mu\
u}{r\omega}+\frac{ \tau_{ll}}{2}\\ \varphi_{z}(\mu_{ij},\
u_{kl},\omega_{pq},r)=-\mu\omega\ln(r-\mu)-\
u\omega\ln( r-\
u)+\mu\
u\text{arctan}\frac{\mu\
u}{r\omega}-r\omega\\ \end{array} \right.