# Dual-Band RF Wireless Power Transfer System with a Shared-Aperture Dual-Band Tx Array Antenna
## 2 Implementation of a Dual-Band RF Wireless Power Transfer (WPT) System
### Design of a Shared-Aperture Dual-Band Tx Array Antenna
\[\beta_{m}=x_{m}\sin\theta^{\prime}\cos\phi^{\prime}+y_{m}\sin\theta^{\prime} \sin\phi^{\prime} \tag{2}\]  
\[\beta_{m}=\frac{2\pi}{\lambda}\bigg{(}\sqrt{\left(x_{m}-x^{\prime}\right)^{2 }+\left(y_{m}-y^{\prime}\right)^{2}+\left(z_{m}-z^{\prime}\right)^{2}}-\sqrt{ \left(x^{\prime}\right)^{2}+\left(y^{\prime}\right)^{2}+\left(z^{\prime} \right)^{2}}\bigg{)} \tag{3}\]  
Equation (4) is the formula used to calculate the transmission efficiency for the RF WPT with the Tx array with \(M\)-elements based on numerical integration [35]. As it is more accurate than the Friis formula [31] in the Fresnel region, the evaluation function of Equation (1) is calculated using this Equation (4). In this paper, given that 32 radiating elements are used in both the 2.4 and 5.8 GHz arrays, \(M\) equals 32 here. \(P_{l_{m}}\) is the excited power to the \(m\)-th Tx element, \(G_{l_{m}}(\theta^{\prime}_{m},\phi^{\prime}_{m})\) is the realized gain of the co-polarization of the \(m\)-th Tx element in the direction of Rx, and \(G_{r}(\theta^{\prime}_{m},-\phi^{\prime}_{m})\) is the realized gain of the co-polarization of the Rx element in the direction of the \(m\)-th Tx element. \(R_{m}\) is expressed as \(\sqrt{\left(x_{m}-x^{\prime}\right)^{2}+\left(y_{m}-y^{\prime}\right)^{2}+ \left(z_{m}-z^{\prime}\right)^{2}}\), indicating the distance between the \(m\)-th Tx element and the Rx. \[\frac{P_{R}}{P_{T}}=\bigg{(}\frac{\lambda}{4\pi}\bigg{)}^{2}\Bigg{|}\frac{ \sum\limits_{m=1}^{M}\sqrt{P_{l_{m}}G_{l_{m}}(\theta^{\prime}_{m},\phi^{\prime }_{m})G_{r}(\theta^{\prime}_{m},-\phi^{\prime}_{m})}\frac{e^{-\left(\beta_{R \text{m}}-\beta_{m}\right)}}{R_{m}}}{\sum\limits_{m=1}^{M}P_{l_{m}}} \tag{4}\]  
During this optimization process, it is ensured that the elements of the two thinned arrays do not overlap and that each thinned array is made to have a quadrant symmetrical geometry around the origin. Optimization of the 2.4 and 5.8 GHz thinned arrays was conducted sequentially in order to maximize Equation (1) based on Equation (4). As a result of the optimization, the final arrangement of each radiating element consisting of the 2.4 and 5.8 GHz thinned arrays on the given single aperture is shown in Figure 2. Here, the 2.4 GHz RHCP patch in Figure 2(a) is applied as the 2.4 GHz radiating element, and the 5.8 GHz RHCP patch in Figure 2(b) is applied as the 5.8 GHz radiating element. These corrugated patches are excited by a coaxial SMA connector and a 1.52 mm thick RF-35 substrate with \(\varepsilon_{r}=3.5\) and \(\tan\delta=0.0018\) is utilized as shown Figure 2(c). Figure 4 illustrates the surface current distribution of the corrugated patch, which is observed from the \(+z\)-direction at the operating frequency. In this figure, \(\bar{I}_{total}\) represents the vector sum of all major current vectors, and \(T\) is the period of the operating frequency. When corrugation on both right and left sides of the square patch is applied and the feeding point is located with offset from the center, \(\bar{I}_{total}\) rotates counterclockwise as the time (\(t\)) increases and the RHCP is achieved. Accordingly, patch elements in the Figure 2(a),b can generate RHCP at 2.482 GHz and at 5.73 GHz, respectively. Figure 5 shows the final configuration and a photograph of the optimized shared-aperture dual-band array. As shown in Figure 2(c), SMA connectors are connected to the patch elements. Figure 6 demonstrates the simulated and measured results of the reflection  
coefficients of several array elements shown in Figure 5. Figure 6a,b describes the reflection coefficients of the 2.4 GHz and 5.8 GHz radiating elements, respectively. Regardless of where the radiating element is located in the final shared-aperture array, the reflection coefficients are less than \(-10\) dB at 2.482 GHz and 5.73 GHz, and the corresponding measured results are in good agreement with the simulated results. Figure 7a shows that the RHCP-realized gains of 2.4 GHz array antenna and 5.8 GHz array antenna in the \(+z\)-direction are approximately 20 dBic and 20.8 dBic, respectively.