# RF Energy Harvesting
## 4 Fundamental Equations  
For every antenna, the emitted field changes its properties based on the distance from the source. In the direct neighbourhood of the antenna, there is an induction zone that is bonded with the current present in the antenna by Maxwell's equations. Further away, there is a Fresnel zone, which is also called the near field. It is a zone in which there is a spherical wave. The amplitude of the wave and the signal strength depend on the square of the distance from the source antenna. Even farther, there is a Fraunhofer region, which is also called the far field. In this zone, the wave is plane and the amplitude of the electromagnetic field decreases with the distance from the source antenna [21]. In this last space, the power received from the transmitter on the antenna of the receiver is calculated using the effective antenna area \(A_{ff}\)[22], as shown in Equation (1).  
\[A_{eff}=\frac{P_{av}}{S} \tag{1}\]  
The symbols used in (1) are as follows:  
* \(A_{eff}\)--antenna's effective area;
* \(P_{av}\)--power delivered to the antenna;
* \(S\)--power density of the wave incident to the antenna.  
\(P_{av}\) is the maximum power that can be received from the antenna and delivered to the energy harvester. It is equal to the real power available in the circuit only when the connection between the antenna and the energy harvesting module is perfectly balanced energetically [22]. This occurs if the load value is equal to the characteristic impedance of the antenna. Figure 1 presents a case wherein the antenna has 50 \(\Omega\) of impedance.  
It can be easily calculated that the maximum possible power transfer from the antenna to the load is equal according to (2).  
\[max(P_{Load})=\frac{V_{ant}^{2}}{4\cdot R_{ant}} \tag{2}\]  
The maximum directivity of each antenna is in relation to its maximum aperture \(A_{eff}\). The aperture of the antenna is the area through which waves travel to and from space [23]. When free space is assumed together with a perfectly matched lossless antenna, the power available in the circuit is described by the Friis equation [22] shown in (3).  
\[P_{av}=A_{eff}S=(\frac{\lambda}{4\pi d})^{2}D_{RX}P_{EIRP} \tag{3}\]  
The coefficients used in Equation (3) are:  
* \(d\)--antenna's directivity;
* \((\frac{\lambda}{4\pi d})^{2}\)--free-space equation;
* \(D_{RX}\)--maximum antenna directivity;
* \(P_{EIRP}\)--effective radiated power.  
Figure 1: Balanced load of the antenna.  
The directivity mentioned in (3) \(d\) describes the relation between the radiation of the antenna in a given direction compared to that of an isotropic antenna.  
The practical appliance of the Friis equation looks as follows:  
\[\frac{P_{r}}{P_{t}}=G_{r}\cdot G_{t}\cdot 10\log_{10}(\frac{\lambda}{4\pi d})^{2} \tag{4}\]  
where the symbols are  
\begin{tabular}{l l} \(P_{r}\) & power at the receiver antenna; \\ \(P_{t}\) & power of the transmitter antenna; \\ \(G_{r}\) & gain of the receiver antenna with respect to an isotropic one; \\ \(G_{t}\) & gain of the transmitter antenna with respect to an isotropic one; \\ \(\lambda\) & longitude of the wave; \\ \(d\) & distance between transmitter and receiver. \\ \end{tabular}  
And the practical logarithmic version expressed in dBm and dBi units is as below:  
\[P_{r}[\text{dBm}]=P_{t}[\text{dBm}]+G_{r}[\text{dBi}]+G_{t}[\text{dBi}]+20 \log_{10}(\frac{\lambda}{4\pi d}) \tag{5}\]  
The power in dBm units is defined as:  
\[P[\text{dBm}]=10\cdot\log_{10}(\frac{P[\text{mW}]}{1\text{ mW}}) \tag{6}\]  
The dBi units express in logarithmic scale the gain of an antenna compared to that of the ideal isotropic antenna. It is assumed that the isotropic antenna has a reference 0 dB gain value.