# Zero Energy IoT Devices in Smart Cities Using RF Energy Harvesting
## 3 Materials and Methods
### Experimental Models
Now we can describe our model as follows: Let \(M_{k}\) be the set of chargers, and \(v_{k}\) be the set of nodes; then, we give the charging model based on Equation (10) as follows:  
\[P_{r}=\frac{\alpha\ P_{t}(M_{k})}{\left(||v_{k}-M_{k}||+\beta\right)^{2}} \tag{11}\]  
where \(\mid\!\!\mid v_{k}-M_{k}\!\mid\!\!\mid\) represent the distance between the device \(\mathrm{v_{k}}\) and charger \(M_{k}\), \(P_{t}\) represents the transmission power of the charger \(M_{k}\), \(\alpha=\frac{G_{t}G_{r}\eta}{Lp}\Big{(}\frac{\lambda}{4\pi}\Big{)}^{2}\). For a directional charging model, the charger and devices are equipped with directional antennas, so the angle of the charger and devices will be kept in mind. Let \(\overset{\rightarrow}{\phi_{mk}}\) be the directional vector of the charger (i.e., charger angle \(\theta_{mk}\)) and \(\overset{\rightarrow}{\phi_{pk}}\) be the directional vector of the device (device angle \(\theta_{pk}\)) then equation 11 can be written for the directional model as follows:  
\[P_{r}(M_{k},v_{kr})=\frac{\alpha\ P_{t}(M_{k})}{\left(||v_{k}-M_{k}||+\beta \right)^{2}} \begin{cases}\frac{||v_{k}-M_{k}||\overset{\rightarrow}{\phi_{ pk}}}{||v_{k}-M_{k}||\overset{\rightarrow}{\phi_{pk}}}&\geq\cos\left(\frac{\theta_{pk} }{2}\right)\\ \frac{||M_{k}-v_{k}||\overset{\rightarrow}{\phi_{mk}}}{||v_{k}-M_{k}||}& \geq\cos\left(\frac{\theta_{mk}}{2}\right)\end{cases}  \tag{12}\]  
Let \(M_{kr}\) be the maximum transmitter range of the energy transmitter and \(v_{kr}\) be the maximum transmission range of the device. Since the energy transmitter has a 0 to 60\({}^{\circ}\) angle, the dipole has a 0 to 360\({}^{\circ}\) angle, and the patch has a 0 to 122 degree range then Equation (12) can be written for a dipole antenna as follows:  
\[P_{r}(M_{k},v_{k})=\frac{\alpha\ P_{t}(M_{k})}{\left(||v_{k}-M_{k}||+\beta \right)^{2}} \begin{cases}\frac{||v_{k}-M_{k}||\overset{\rightarrow}{\phi_{ pk}}}{||v_{k}-M_{k}||\overset{\rightarrow}{\phi_{pk}}}&\geq\cos\left(\frac{ \theta_{pk}}{2}\right)\text{ where }0\ \leq\theta_{pk}\leq 2\pi\\ \frac{||M_{k}-v_{k}||\overset{\rightarrow}{\phi_{mk}}}{||v_{k}-M_{k}||}& \geq\cos\left(\frac{\theta_{mk}}{2}\right)\text{ where }0\ \leq\theta_{mk}\leq 60\end{cases}  \tag{13}\]  
Equation (12) can be written for a patch antenna as follows:  
\[P_{r}(M_{k},v_{k})=\frac{\alpha\ P_{t}(M_{k})}{\left(||v_{k}-M_{k}||+\beta \right)^{2}}\begin{cases}\frac{||v_{k}-M_{k}||\overset{\rightarrow}{\phi_{ pk}}}{||v_{k}-M_{k}||\overset{\rightarrow}{\phi_{pk}}}&\geq\cos\left(\frac{ \theta_{pk}}{2}\right)\text{ where }0\ \leq\theta_{pk}\leq 122\\ \frac{||M_{k}-v_{k}||\overset{\rightarrow}{\phi_{mk}}}{||v_{k}-M_Currently, we have two sensors and one charger, so we can create fixed points for sensors when the charger is moving. The Powercast transmitter directionally transmits the energy up to a 60-degree area. The evaluation board receives this signal and converts it to DC. If the evaluation board is not lying in the direction of the transmitter, it cannot harvest energy. Consequently, it halts all processes and leads to a dead situation. Next, the Powercast Company provides a transmitter range of up to 80 feet [29], which means that the farthest node can be placed 80 feet away from the transmitter. So, the maximum range is from 0 to 80 feet. Here, we can define two problems for the energy transmitter: distance and range. If the distance is 80 feet above then the receiver antenna (device) cannot receive energy, and if the evaluation board took place outside of a 60-degree area, then the receiver antenna cannot recieve the energy. We need a solution that meets the above problems efficiently and makes the network nearly perpetual.