## 2 Electromagnetic Energy Harvester: Principles and Design
### Modeling of the Energy Harvester  
In this section, the analysis of the electromagnetic energy harvester and the calculation of the magnetic flux density are described. The magnetic nodes theory in [27; 28; 29] indicates that a magnetic field can be calculated by eight nodes of coordinates. The spatial component of the magnetic field at point \(P\) (\(r\), \(\theta\), \(z\)) is:  
\[B_{z}(r,\theta,z)=\sum_{k=1}^{8}\frac{\varepsilon_{k}}{\pi}\text{arctan} \left(\frac{\delta_{k}\sigma_{k}}{dz_{z}}\right) \tag{1}\]  
where \(\delta_{k}\), one spatial component of the distance in cartesian coordinates between point P and the node can be expressed as \(r\cos\theta-r_{k}\cos\theta_{k}\), \(\sigma_{k}\), another spatial component of the distance in cartesian coordinates between point \(P\) and the node can be expressed as \(r\sin\theta-r_{k}\sin\theta_{k}\), \(d_{z}\), the third spatial component of the distance between point \(P\) and the node can be expressed as \(z-z_{k}\), \(d\), the spatial distance between point \(P\) and the  
Figure 1: Schematic of the proposed REMEH with coils and magnets. (**a**) Electromagnetic energy harvesting structure; (**b**) Halbach array; (**c**) two-directional magnet array.  
node, can be expressed as \(\sqrt{\left(r\cos\theta-r_{k}\cos\theta_{k}\right)^{2}+\left(r\sin\theta-r_{k}\sin \theta_{k}\right)^{2}+\left(z-z_{k}\right)^{2}}\), and \(\varepsilon_{k}\) is the magnetization of each node. The values of the magnetization of adjacent nodes are the opposite.  
A circular magnet rotates around the \(Z\) axis. The magnetic flux in the \(Z\) direction of a turn of the coil is:  
\[\phi(t)=\int\limits_{r_{1}}^{r_{2}}\int\limits_{0}^{\pi/8}B_{z}(r,\beta(t)+ \alpha,z)rdadr \tag{2}\]  
where \(r_{1}\) and \(r_{2}\) are the inner radius and the outer radius of the circular magnet array, respectively, the gap between the magnet array and the coils is described as \(d_{m}\), \(z_{k}\) is the width of the coils, and \(\beta(t)\) is the angular displacement between coil \(i\) and the magnet as shown in Figure 2. The induced voltage in coil \(i\) is caused by [PERSON]'s law [30] and can be indicated as:  
\[\begin{array}{l}U_{i}(t)=\int\limits_{d_{m}}^{d_{m}+z_{k}}-\frac{d\phi(t)}{ dt}n_{z}dz=\int\limits_{d_{m}}^{d_{m}+z_{k}}-\frac{d\phi(t)}{d\beta(t)}\frac{d \beta(t)}{dt}n_{z}dz\\ =-\omega\int\limits_{d_{m}}^{d_{m}+z_{k}}\int\limits_{r_{1}}^{r_{2}}\frac{\pi /8}{\alpha}n_{z}\frac{\partial B_{z}(r,\beta(t)+\alpha,z)}{\partial\beta}rdadrdz \\ =-n_{z}\omega\int\limits_{d_{m}}^{d_{m}+z_{k}}\int\limits_{r_{1}}^{r_{2}}B_{z} (r,\beta(t)+\alpha,z)\left|\begin{array}{cc}\pi/8&rdrdz\\ 0&rdrdz\end{array}\right.\\ =-n_{z}\omega\int\limits_{d_{m}}^{d_{m}+z_{k}}\int\limits_{r_{1}}^{r_{2}} \Delta B_{z}drdrdz\end{array} \tag{3}\]  
where \(n_{z}\) is the number of coil turns per unit of height. It can be seen from formula (3) that the energy conversion efficiency can be judged by \(\Delta\)B at the coil when the coil density, rotation speed, and radius parameters are all fixed for a certain rotating electromagnetic energy harvester.  
Figure 2: Dynamic state of an energy harvester with multiple parameters: (**a**) top view of the magnets and coil with the Halbach array; (**b**) top view of the two-directional magnet array and coil; (**c**) three-dimensional view of the magnets and coil with the Halbach array in a dynamic response state; (**d**) three-dimensional view of cylindrical coordinates; (**e**) three-dimensional view of the magnets and coil with the two-directional array in a dynamic response state.