## 2 Frequency Reconfigurable Microstrip Patch Antenna Array  
### The Configuration and Design of the Patch Antenna Array  
A conventional antenna array only works at one frequency. This is because when the conventional antenna is used as the transmitter for wireless power transfer (WPT), the conventional antenna used as the receiver only receives the energy from the transmitter at the same working frequency. Also, when the working frequency of the receiver changes, the power transfer efficiency of the system will reduce. In addition to the conventional antenna only being able to power the receiver at one working frequency, it cannot power the receiver for different frequencies, regardless of energy demands. This may cause malfunctions and failures, and the PTE of the system may decrease as a result. The frequency reconfigurable antenna continuously adjusts the working frequency within a certain frequency range. The WPT system based on a frequency reconfigurable antenna can power the receivers at different frequencies. It was shown that the PTE of the WPT system remains stable at different frequencies. Also, the PTE of the system does not clearly decrease as the resonant frequency of the receiver changes. Therefore, the frequency reconfigurable array more important than the conventional antenna array for wireless power transfer. The operating mechanism of the array is that the array radiates the electromagnetic wave if four patches are excited. Under the excitation of the main mode, the direction of the electric field changes along the length of the half-wavelength patch. The antenna array radiates energy mainly through the gap between the open edge of the patch and the floor, and the electromagnetic wave radiated over the narrow seam can be equivalent to the magnetic flux. According to the array antenna theory, the radiation field at the center of two arrays is the largest, as the direction of patch normal, while the radiation field deviating from the corresponding direction with the normal direction becomes smaller. The electric field of the two radiated edges can be decomposed into the vertical and horizontal components relative to the ground plane. The length of the radiation patch is set as a half wavelength. According to the transmission line theory of half wavelengths, the vertical components of the two radiation edge crevice decomposition cancel each other out in the far field and generate no radiation, while the horizontal component has the same direction and produces the strongest radiation, meaning that the radiation field is the strongest in the direction perpendicular to the antenna surface. The frequency reconfigurable antenna array is fed by a coaxial feed. The feed network of the antenna array uses the power divider. The power divider is designed to provide the same excitation power. For a frequency tuning operation, varactor diodes are used. The varactor diode is placed between the larger main patch and the smaller side patch, which has the variable bias voltage provided by the adjustable DC power source. By changing the capacitance of those varactors, the resonance frequency of the patch antenna array can be tuned. The antenna dimensions can be derived from the working frequency of the microstrip patch antenna. The variable symbol \(W\) represents the width of the patch, which can be derived as follows [28]:  
\[W=\frac{c}{2f}(\frac{\varepsilon_{r}+1}{2})^{-\frac{1}{2}} \tag{1}\]  
where the variable symbol \(c\) is the speed of light, and the variable symbol \(\varepsilon_{r}\) is the relative dielectric constant of the dielectric material. The dielectric material is epoxy resin, while the variable symbol \(f\) is the working frequency of the patch antenna. The thickness of dielectric material is \(h\).