# A Power Processing Circuit for Indoor Wi-Fi Energy Harvesting for Ultra-Low Power Wireless Sensors
## 3 Analytical Analysis
### Matched and Unmatched Antenna-Rectifier Circuit  
The incident power dictates the output voltage of the matching circuit at the receiving antenna and the input impedance of the rectifying circuit. The two different matching conditions are considered; the conditions are when the antenna-rectifier is matched, and another condition is when the antenna-rectifier is unmatched. For the matched condition, the matching network as shown in Figure 4 is considered. To make sure the matching circuit conjugately matches the antenna to the rectifying circuit, the input impedance of the matching circuit needs to be equal to the antenna impedance (\(Z_{in}=R_{g}\)) or \(Y_{in}R_{g}=1\). Since the matching circuit is assumed lossless, the output power is assumed to be equal to the input power, \(P_{out}=P_{in}\). The antenna-rectifying voltage gain can be written as  
\[A_{rf}=\frac{v_{rf}}{v_{in}}=\frac{1}{2}\sqrt{\frac{\Re\{Y_{in}\}}{\Re\{Y_{L }\}}} \tag{4}\]  
From Figure 4, the input impedance \(Z_{in}\) for the antenna-rectifier matched condition can be calculated by  
\[Z_{in}=jX_{1}+\frac{1}{G_{L}+j(B_{1}+B_{L})}=\frac{G_{L}}{G_{L}^{2}+\left(B_{ 1}+B_{L}\right)^{2}}-j\frac{\left(B_{1}+B_{L}\right)}{G_{L}^{2}+\left(B_{1}+B _{L}\right)^{2}}+jX_{1} \tag{5}\]  
For the input impedance to be equal to \(R_{g}\), the following two conditions must be applied which are:  
\[R_{g}=\frac{G_{L}}{G_{L}^{2}+\left(B_{1}+B_{L}\right)^{2}};X_{1}=\frac{\left( B_{1}+B_{2}+B_{L}\right)}{G_{L}^{2}+\left(B_{1}+B_{L}\right)^{2}} \tag{6}\]  
The real part of the input admittance is given by  
\[\Re\{Y_{in}\}=\frac{G_{L}^{2}+\left(B_{1}+B_{L}\right)^{2}}{G_{L}} \tag{7}\]  
The circuit voltage gain for the antenna-rectifier matched condition is given by  
\[A_{rf}=\frac{V_{rf}}{V_{in}}=\frac{1}{2}\sqrt{1+\left(\frac{B_{1}+B_{L}}{G_{ L}}\right)^{2}}=\frac{1}{2}\sqrt{1+Q^{2}} \tag{8}\]  
where \(\left(\left(B_{1}+B_{L}\right)\right)/G_{L}\) is the quality factor at the output of the matching circuit. As defined in (6), the rectifier/switching circuit defines parameter \(G_{L}\) and \(B_{L}\). By proper choice of \(B_{1}\), the output voltage can be decided, and the value of \(R_{g}\) and \(X_{1}\) are calculated using (9). Indirectly, it is assumed that \(v_{ac}\) and \(R_{g}\) are chosen individually. Moreover, these two values are interrelated through the power available from the harvesting antenna, \(P_{inc}\).  
\[P_{inc}=\frac{v_{ac}^{2}}{8R_{g}} \tag{9}\]  
To calculate the output voltage at the end of the matching circuit, (6) and (9) are substituted into (10) to give the value of;  
\[v_{rf}=G_{v}v_{ac}=\sqrt{\frac{2P_{inc}}{G_{L}}} \tag{10}\]  
From (10), the voltage at the input of the rectifier/switching is consequently emitted by the available power from the harvesting antenna and the real part of the input admittance of the rectifier/switching circuit. Therefore, to maximise \(v_{\textit{rf}}\), the real admittance of the rectifier/switching circuit needs to be as low as possible. For the condition when the impedance of the harvesting antenna and the rectifier circuit is unmatched, this gives \(Y_{in}R_{g}\
eq 1\) and  
\[\left|\frac{1}{1+Y_{in}R_{g}}\right|=\frac{1}{\sqrt{\left(1+G_{in}R_{g}\right) ^{2}+B_{in}^{2}R_{g}^{2}}} \tag{11}\]  
where  
\[Y_{in}=\Re\{Y_{in}\}+j\Im\{Y_{in}\}=G_{in}+jB_{in} \tag{12}\]  
The matching circuit output voltage, \(v_{\textit{rf}}\), at this condition is written as  
\[v_{\textit{rf}}=\sqrt{\frac{2P_{inc}}{G_{L}}}2\sqrt{\frac{G_{in}R_{g}}{\left( 1+G_{in}R_{g}\right)^{2}+B_{in}^{2}R_{g}^{2}}} \tag{13}\]  
The output voltage of the matching circuit is identical to the output voltage for the matched situation multiplied by a multiplicative factor. This factor is equal to 1 for the case when the value of \(G_{in}=1/R_{g}\) and \(B_{in}=0\) (matched condition) is smaller than 1 for all situations. Hence, for the voltage to reach a maximum, the harvesting antenna should be impedance matched.