# Wireless Power Transfer System Based on Strapping Resonators
## 2 Analysis of the Strapped Resonator  
The WPT system based on strapped resonators (SR) is shown in Figure 1a. The system consists of two strapped resonators. The distance between the source resonator and load resonator is \(l\). One coaxial probe is used to excite the source resonator, while another one is used to receive the power from the load resonator. Figure 1b shows the structure of the SR. The structure of the SR consists of three parts. The first part is a metal shell. The second part consists of four metal vanes that divide the interior of the structure into four sub-cavities. The third part consists of two pairs of strapped rings, which connect the four vanes alternately. The strapped ring can improve the mode separation so that the resonant frequency of fundamental mode is far away from the resonant frequency of high-order mode. Figure 1c shows the internal structure of a single strapped resonator. The details of the single SR are shown in Table 1.  
Figure 1: Wireless power transfer (WPT) system based on strapped resonator (SR): (**a**) Coupling model, (**b**) structure of strapped resonator, and (**c**) details of the strapped resonator. The resonance characteristics of the strapped resonator are investigated using numerical simulation. Eigen-mode simulations are carried out by commercial software CST (Computer Simulation Technology) Microwave Studio. In the simulation model, to calculate the unloaded quality factor \(Q_{0}\), the material of the SR is set to be oxygen-free copper (with electrical conductivity of 3.56e + 007 S/m). The frequency of the fundamental resonant mode is 0.65 GHz. The field distribution of the fundamental mode is shown in Figure 2. As shown in Figure 2a,b, in the adjacent sub-cavities, the fields have the same amplitude but opposite direction. The phase difference of the oscillations in the two adjacent sub-cavities is \(\pi\). Thus, the fundamental mode is named as \(\pi\) mode. Figure 2c,d illustrate the electric and magnetic fields from longitudinal view. The magnetic field distributes away from the center of the structure and its direction is axial. In contrast, the electric field distribution is closer to the center of the structure and its direction is radial. Therefore, with proper excitation, the strapped resonator can exhibit a strong \(\pi\) mode magnetic resonance. The field \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Unit** & **Value** & **Description** \\ \hline \(\theta\) & degree & 35 & aperture angel of the vane \\ \(h\) & mm & 40 & height of the vane \\ \(h_{1}\) & mm & 3 & width of the strapped ring \\ \(d_{1}\) & mm & 28 & inner diameter of the SR \\ \(d_{2}\) & mm & 32 & length of the vane \\ \(d_{3}\) & mm & 16 & position of the strapped ring connecting point \\ \(d_{4}\) & mm & 8.5 & position of the coaxial connecting point \\ \(l\) & mm & 35 & distance between source resonator and load resonator \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimension parameters of the strapped resonator. SR: strapped resonator. Figure 2: Field distributions of the \(\pi\) mode: (**a**) Electric field distribution from the cross-sectional view, (**b**) magnetic field distribution from the cross-sectional view, (**c**) electric field distribution from the longitudinal section view, and (**d**) magnetic field distribution from the longitudinal section view. distribution characteristics of the \(\pi\) mode are suitable for transferring power from one resonator to another through magnetic coupling. Figure 3 illustrates the electromagnetic field distributions of the first higher mode (\(\pi+1\) mode) from the longitudinal section view. The \(\pi+1\) mode field between two adjacent cavities is continuous. The directions of the magnetic fields in adjacent cavities are the same. Unlike the \(\pi\) mode, the \(\pi+1\) mode does not exhibit significant magnetic resonance or electrical resonance. When the system operates at \(\pi+1\) mode, mixed coupling will play the major role in wireless power transfer.